A  COURSE  OF  INSTRUCTION 


IN  THE 


GENERAL   PRINCIPLES 


OF 


CHEMISTRY 


BY 

ARTHUR  A.  NOYES 

AND 

MILES  S.  SHERRILL 


PRINTED   IN   PRELIMINARY  FORM  FOR   THE  CLASSES  OF  THE 
MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY 


BOSTON : 

THOMAS  TODD  Co.,  PRINTERS 
1917 


COPYRIGHT,  191? 
BY  ARTHUR  A.  NOYES 
AND  MILES  S.  SHERRILL 


CHAPTER   I 
THE  COMPOSITION  OF  SUBSTANCES 


1.  Definition  of  the  Field  of  Chemistry.    Its  General  Principles 
the  Subject  of  this  Course. —  Chemistry  treats  of  the  composition  of 
substances,  of  their  properties  in  relation  to  their  composition,  of 
changes   in   their   composition,    and   of   the   effects    attending    such 
changes. 

General  chemistry,  often  called  also  theoretical  or  physical  chem- 
istry, treats  of  the  general  principles  which  have  been  found  to  ex- 
press certain  common  characteristics  of  the  numerous  phenomena 
of  chemistry.  To  a  discussion  of  the  more  important  of  these  general 
principles  this  Course  will  be  devoted.  The  divisions  of  the  subject 
will  be  taken  up  in  the  order  in  which  they  were  named  in  the  above- 
given  definition  of  the  field  of  chemistry. 

2.  Pure  Substances  and  Mixtures,  and  the  Law  of  Definite  Propor- 
tions.—  Out  of  the  materials  occurring  in  nature  there  can  be  prepared 
substances  which,  when  subjected  to  suitable  processes  of  fractionation 
(that  is,  to  operations  which  resolve  the  materials  into  parts  or  frac- 
tions),  always  yield  fractions   whose  properties   are  identical  when 
measured  at  the  same  pressure  and  temperature.   Such  substances  are 
called  pure  substances;  other  substances  which  can  be  resolved  by  such 
processes  into  fractions  with  different  properties  being  called  mixtures. 
For  example,  whether  a  solid  material  is  a  pure  substance  or  mixture 
may  be  determined  by  partially  melting  or  vaporizing  it  or  by  partially 
dissolving  it  in  solvents,  and  by  comparing  the  value  of  the  density,, 
melting-point,  or  some  other  sensitive  property,  of  the  unmelted,  un- 
vaporized,  or  undissolved  part  with  that  of  the  original  material. 

The  fundamental  idea  involved  in  the  preceding  considerations  is 
that  there  exists  an  order  of  substances,  called  pure  substances,  of  rela- 
tively great  stability  toward  resolving  agencies,  each  one  of  which  has 
a  perfectly  definite  set  of  properties,  sharply  differentiated  from  those 
of  other  pure  substances;  so  that  there  is  not  a  continuous  series  of 
pure  substances  whose  properties  pass  over  into  one  another  by  insensi- 
ble gradations. 

1 


I  CMfO&ITION  OF  SUBSTANCES 


This  principle  of  definiteness  of  properties  in  general  applies  also 
to  the  elementary  composition  of  pure  substances.  This  fact  is  ex- 
pressed by  the  law  of  definite  proportions,  which  states  that  a  pure  sub- 
stance, however  it  be  prepared,  always  contains  its  elements  in  exactly 
the  same  proportions  by  weight. 

3.  The  Law  of  Combining  Weights.  —  To  the  various   elements 
definite  numerical  values  can  be  assigned  which  accurately  express 
the  weights   of  them,    or  small  multiples   of   the  weights  of   them, 
which  are  combined  with  one  another  in  all  pure  substances.  Such 
numerical  values  are  called  the  combining  weights  of  the  elements. 
They  are  essentially  relative  quantities.  Adopting  16  as  the  combining 
weight  of  oxygen,  the  combining  weight  of  any  other  element  may  be 
defined  to  be  that  weight  of  it  which  combines  with  16  parts  of  oxygen, 
or  with  some  small  multiple  or  submultiple  of  16  parts  of  oxygen  ;  and 
the  above  principle,  known  as  the  law  of  combining  weightsf  can  be 
expressed  as  follows  :  Elements  are  present  in  pure  substances  only  in 
the  proportions  of  their  combining  weights  or  of  small  multiples  of 
them. 

Prod.  1.  The  oxide  of  a  certain  element  contains  30.06%  of  oxygen. 
and  the  sulphide  of  the  same  element  contains  53.46%  of  sulphur.  What 
does  the  law  of  combining  weights  show  as  to  the  relative  weights  of 
oxygen  and  sulphur  that  may  be  present  in  the  pure  compounds  of  these 
two  elements? 

4.  Determination  of  Combining  Weights.  —  The  way  in  which  some 
important  combining  weights  have  been  determined  is  illustrated  by 
the  following  problem. 

Pro&.  2.  Determine  the  exact  combining  weights  of  silver,  potassium, 
and  chlorine  from  the  following  data  :  In  a  series  of  eight  experiments 
801.48  g.  of  pure  potassium  chlorate  were  ignited  or  treated  with  hydro- 
chloric acid,  yielding  485.66  g.  of  potassium  chloride.  In  another  series  of 
five  experiments  24.452  g.  of  pure  potassium  chloride  were  dissolved  in 
water  and  precipitated  with  silver  nitrate,  whereby  47.013  g.  of  silver 
chloride  were  obtained.  In  a  third  series  of  ten  experiments  S2.669  g.  of 
pure  silver  were  dissolved  in  nitric  acid  and  precipitated  with  hydro- 
chloric acid,  yielding  109.840  g.  of  silver  chloride. 

The  combining  weights  adopted  by  the  International  Committee 
on  Atomic  Weights  for  the  more  important  elements  are  presented  in 
the  following  table,  those  multiples  being  given  which  have  been  shown 
to  be  the  atomic  weights,  as  described  in  Art.  17.  The  table  is  therefore 
also  one  of  atomic  weights. 


THE  COMPOSITION  OF  SUBSTANCES 


3 


Aluminum     .     .     .     .  Al  27.1 

Antimony      .     .     .     .  Sb  120.2 

Argoii A  39.88 

Arsenic As  74.96 

Barium Ba  137.37 

Beryllium      ....  Be        9.1 

Bismuth Bi  208.0 

Boron B       11.0 

Bromine Br  79.92 

Cadmium      .     .     .     .  Cd  112.40 

Calcium Ca  40.07 

Carbon C  12.005 

Chlorine 01  35.46 

Chromium     .     .     .     .  Cr  52.0 

Cobalt Co  58.97 

Copper Cu  63.57 

Fluorine F  19.0 

Gold Au  197.2 

Helium He        4.00 

Hydrogen       .     .     .     .  H         1.008 

Iodine  .  I  126.92 


Iron Fe  55.84 

Lead Pb  207.20 

Lithium Li  6.94 

Magnesium   ....  Mg  24.32 

Manganese    ....  Mu  54.93 

Mercury Hg  200.6 

Nickel Ni  58.68 

Nitrogen N  14.01 

Oxygen     .....  O  16.00 

Phosphorus   .     .     .     .  P  31.04 

Platinum Pt  195.2 

Potassium     .......  K  39.10 

Radium     .     .     ,     ..    .  Ra  226.0 

Silicon      .....  Si  28.3 

Silver Ag  107.88 

Sodium Na  23.00 

Strontium     .     .     .     .  Sr  87.63 

Sulphur S  32.06 

Thallium Tl  204.0 

Tin Sn  118.7 

Zinc                               ,  Zn  65.37 


5.  The  Atomic  and  Molecular  Theories  account  for  the  fact 
that  elements  combine  with  one  another  only  in  the  proportions  of 
their  combining  weights  or  small  multiples  of  them  by  assuming 
that  any  mass  of  each  element  is  made  up  of  a  very  large  number 
of  extremely  small  particles  called  atoms;  that  these  are  exactly 
alike  in  every  respect;  that  they  are  not  subdivided  by  chemical 
processes;  that  there  are  as  many  kinds  of  atoms  as  there  are  ele- 
ments; that  the  atoms  associate  with  one  another,  usually  in  small 
numbers,  forming  a  new  order  of  distinct  particles  called  molecules; 
that  pure  chemical  substances  are  made  up  of  only  one  kind  of  mole- 
cules, while  mixtures  contain  two  or  more  kinds;  and  that  the 
molecules  of  elementary  substances  consist  of  atoms  of  the  same 
kind,  those  of  compound  substances  of  atoms  of  different  kinds. 

These  assumptions  in  regard  to  atoms  and  molecules  have  now 
been  confirmed  in  so  many  ways  that  they  are  no  longer  hypothetical. 

By  the  above  statement  chemical  substances  are  implicitly  defined 
from  the  molecular  standpoint  as  pure  substances  which  contain  only  a 
single  kind  of  molecule.  Thus,  the  pure  substance  liquid  water  contains 
the  two  chemical  substances  whose  molecules  are  H2O  and  H4O2,  these 
being  always  present  in  definite  proportions  at  any  definite  temperature 


4  THE  COMPOSITION  OF  SUBSTANCES 

and  pressure,  since  equilibrium  is  instantaneously  established  between 
them ;  but  the  pure  substance  water-vapor,  which  contains  only  mole- 
cules of  the  form  H2O,  is  a  single  chemical  substance.  Pure  substances 
which,  like  water,  water-vapor,  and  ice,  are  converted  into  each  other 
by  changes  of  pressure  and  temperature,  are  commonly  spoken  of  as 
the  same  substance;  but  they  may  consist  of  different  chemical  sub- 
stances, as  has  been  just  illustrated. 

The  relative  weights  of  the  atoms  of  the  various  elements  and 
of  the  molecules  of  the  various  substances  are  called  their  atomic 
weights  and  molecular  weights,  respectively;  and  as  a  standard  of 
reference,  the  weight  of  the  oxygen  atom  taken  as  16  is  adopted. 

The  atomic  theory  evidently  requires  that  the  weights  of  elements 
that  combine  with  one  another  be  proportional  to  the  weights  of  their 
atoms  or  to  small  multiples  of  those  weights;  in  other  words,  that  the 
atomic  weights  be  equal  to  the  combining  weights  or  to  small  multiples 
of  them.  Which  multiple  of  the  combining  weight  is  the  atomic  weight 
cannot  be  derived  from  the  elementary  composition  of  substances.  It 
can,  however,  be  derived  from  other  properties,  with  the  aid  of  certain 
other  principles. 

6.  Chemical  Formulas,  Formula- Weights,  and  Equivalent- Weights. 
— In  order  to  express  the  gravimetric  composition  of  compounds,  the 
symbols  of  the  elements  are  considered  to  represent  their  atomic 
weights  and  are  written  in  sequence  with  such  integers  as  subscripts 
as  will  make  the  resulting  formula  express  the  proportions  by  weight 
of  the  elements  in  the  compound. 

The  formula  is  commonly  so  written  as  to  represent  also  the  number 
of  atoms  of  each  element  present  in  the  molecule,  when  this  has  been 
determined  (by  any  of  the  methods  described  in  later  articles). 

The  formula  represents,  in  addition,  a  definite  weight  of  tho 
substance,  namely,  the  weight  in  grams  which  is  equal  to  the  sum 
of  the  numbers  represented  by  the  symbols  of  the  elements  in  the 
formulas.  This  weight  is  called  the  formula-weight  of  the  substance. 
Thus  the  formula  HNO3  denotes  1.008  -f- 14.01  -f  (3  X  16.00)  or  63.02 
grams  of  nitric  acid. 

Those  weights  of  various  substances  which  enter  into  chemical 
reactions  with  one  another  are  called  equivalent  weights.  Adopting 
one  formula-weight  or  1.008  grams  of  the  element  hydrogen  as  the 


THE  COMPOSITION  OF  SUBSTANCES  5 

standard  of  reference,  the  equivalent-weight  or  one  equivalent  of 
any  substance  is  denned  to  be  that  weight  of  it  which  reacts  with 
this  standard  weight  of  hydrogen,  or  with  that  weight  of  any  other 
substance  which  itself  reacts  with  this  standard  weight  of  hydrogen. 
Thus  the  equivalent-weight  of  each  of  the  following  substances  is 
that  fraction  of  its  formula-weight  which  is  indicated  by  the  coeffi- 
cient preceding  the  formula:  £C12;  |O2;  lAg;  £Zn;  £Bi;  INaOH; 
£Ba(OH)2;  £H2SO4;  £H3P04;  JAlCi,;  £K4Fe(CN)6.  The  equiva- 
lent-weight of  a  substance  may  have  different  values  depending  on 
whether  it  is  considered  with  reference  to  a  reaction  of  metathesis 
or  to  one  of  oxidation  and  reduction.  Thus,  the  metathetical  equiva- 
lent of  ferric  chloride  is  ^FeCL,,  but  its  oxidation-equivalent  (with 
respect  to  its  conversion  to  ferrous  chloride)  is  !FeCl3;  the  metathet- 
ical equivalent  of  potassium  chlorate  is  1KC1O3,  but  its  oxidation- 
equivalent  (with  reference  to  its  reduction  to  KOI)  is 


CHAPTER   II 

PROPERTIES  OF  GASES  RELATED  TO  MOLECULAR 
COMPOSITION 


PRESSURE-VOLUME    RELATIONS    OF    GASES 

7.  The  Laws  of  Perfect  Gases  are  the  limiting  laws  to  which 
gases  conform  more  and  more  closely  as  their  pressure  approaches 
zero.    These  laws  express  fairly  closely  (within  less  than  one  percent) 
the  behavior  of  gases  up  to  pressures  not  much  greater  than  one 
atmosphere,  provided  they  are  far  removed  from  their  temperatures 
of  condensation. 

8.  Boyle's   Law. — At   any   definite  temperature  the  pressure   p 
of  a  definite  weight  m  of  any  perfect  gas  is  inversely  proportional 
to  its  volume  v.  Or,  since  density  d  is  defined  to  be  the  ratio  m/v  of  the 
weight  of  a  substance  to  its  volume,  the  pressure  of  any  perfect  gas  at 
any  definite  temperature  is  directly  proportional  to  its  density. 

9.  Gay-Lussac's   Law   of   Temperature-Effect. — When   different 
perfect  gases  are  heated  or  cooled  from  a  definite  initial  to  a  definite 
final  temperature,  their  pressure-volume  products  p  v  change  by  the 
same  fractional  amount. 

This  principle  and  its  degree  of  accuracy  when  applied  to  gases 
at  moderate  pressures  are  illustrated  by  the  following  data:  When 
1000  ccm.  of  gas  at  0°  and  a  pressure  of  1000  mm.  of  mercury 
are  heated  to  100°  at  constant  pressure,  the  volume  increases  by 
367  ccm.  with  nitrogen,  366  ccm.  with  hydrogen,  and  374  ccm.  with 
carbon  dioxide;  when  heated  at  constant  volume,  the  pressure  in- 
creases by  367,  366,  and  373  mm.,  with  the  three  gases,  respectively. 

10.  Definition  of  Absolute  Temperature. — Absolute  temperature 
T  is  so  defined  that  the  pressure-volume  product  p  v  of  a  perfect 
gas  is  directly  proportional  to  it.    Since  the  pressure-volume  product 
decreases    -—^    of  its  value  at  0°  for  each  degree  that  the  tempera- 
ture decreases,  the  absolute  zero  is  at  — 273.1° ;   and  the  absolute 
temperature  T  is  equal  to  t  -f-  273.1  (approximately  273*),  where  i 
is  the  ordinary  centigrade  temperature. 

*Approximate  values  of  numerical  constants  which  it  is  well  to  remember  are 
printed  in  black-face  type.  These  values  will  be  given  so  as  to  be  accurate  to  0.1%. 

6 


PRESSURE-VOLUME  RELATIONS  OF  GASES        7 

11.  Expression  of  the  Physical  Laws  of  Perfect  Gases. — The  laws 
of  Boyle  and  Gay-Lussac,  the  definition  of  absolute  temperature,  and 
the  obvious  proportionality  between  the  value  of  the  pressure-volume 
product  and  the  weight  ra  of  the  gas  are  all  expressed  by  the  equation 
p  i}  =  m  R  T,    in  which  R  is  a  constant,  evidently  representing  the 
value  of  pv/T  for  one  gram  of  the  gas,  which  has  different  values  for 
different  gases. 

Pro 6.  1.  From  the  fact  that  the  density  of  oxygen  is  0.001429  g.  per 
ccm.  at  0°  and  1  atm.  calculate  the  volume  in  liters  of  32  g.  of  it  at  20° 
and  1  atm. 

12.  Law  of  Combining  Volumes  and  the  Principle  of  Avogairo. — 

The  foregoing  physical  laws  acquire  an  important  chemical  significance 
by  reason  of  the  law  of  combining  volumes,  which  may  be  stated  as 
follows :  Those  quantities  of  perfect  gases  that  are  involved  in  chem- 
ical reactions  with  one  another  have  at  the  same  temperature  and 
pressure  volumes  which  are  equal  or  small  multiples  of  one  another. 
Thus  the  quantities  of  hydrogen  and  of  oxygen  which  unite  with  each 
other  to  form  water  have  volumes  whose  ratio  approaches  the  exact 
value  2 :1  as  the  pressure  of  the  gases  approaches  zero.  In  other  words, 
those  quantities  of  different  substances  which  as  perfect  gases  have  the 
same  value  of  the  product  pv/T  are  equal  to,  or  are  small  multiples  of, 
the  quantities  which  are  involved  in  chemical  reactions  with  one  an- 
other. Now,  since  according  to  the  molecular  theory  substances  react 
by  molecules,  these  quantities  of  different  perfect  gases  which  have  the 
same  value  of  pv/T  must  contain  either  an  equal  number  of  molecules 
or  small  multiples  of  an  equal  number. 

These  considerations  suggest  a  simpler  hypothesis,  known  as  the 
principle  of  Avogadro,  which  may  be  stated  as  follows:  Those  quanti- 
ties of  any  perfect  gases  which  have  the  same  volume  at  the  same  tem- 
perature and  pressure,  and  therefore  the  same  value  of  pv/T  at  any 
temperature  and  pressure,  contain  an  equal  number  of  molecules. 

This  principle,  originally  hypothetical,  has  now  been  so  fully  veri- 
fied that  it  has  become  one  of  the  fundamental  laws  of  chemistry. 

Prob.  2.  From  the  principle  of  Avogadro  show  that  the  densities  of 
perfect  gases  at  the  same  temperature  and  pressure  are  proportional  to 
their  molecular  weights. 

13.  Empirical  Definition  of  Molecular  Weight  and  of  Mol.— The 

principle  of  Avogadro  evidently  enables  the  relative  weights  of  the 


8  MOLAL  PROPERTIES  OF  GASES 

molecules  of  different  gaseous  substances  to  be  determined.  To  express 
these  relative  weights  more  definitely,  the  ratio  of  the  weight  of  the 
molecule  of  any  substance  to  the  weight  of  the  molecule  of  oxygen 
taken  as  32  is  commonly  considered,  this  ratio  being  called  the  molecu- 
lar weight  M  of  the  substance.  The  number  32  is  adopted  as  the  refer- 
ence quantity  of  oxygen,  since  (as  shown  in  Prob.  13)  it  corresponds 
to  the  adoption  of  16  as  the  atomic  weight  of  oxygen. 

The  so-defined  molecular  weight  of  a  substance  may  evidently  be 
experimentally  determined  by  finding  the  number  of  grams  of  it  which 
have  that  value  of  pv/T  which  32  grams  of  oxygen  have,  when  both 
substances  are  in  the  state  of  perfect  gases.  This  number  of  grams  is 
called  one  mol  of  the  substance — a  unit-quantity*  which  has  great  im- 
portance in  chemical  considerations,  both  because  it  is  directly  related 
to  molecular  weight  and  because  it  makes  possible  a  general  expression 
of  the  laws  of  perfect  gases,  as  shown  in  the  following  Article. 

14.  General  Expression  of  the  Laws  of  Perfect  Gases. — Represent- 
ing by  R  the  constant  value  of  pv/T  for  one  mol  and  by  N  the  number 
of  mols  of  the  gas  present,  the  laws  of  perfect  gases  may  be  expressed 
in  a  general  form  by  the  following  equation,  hereafter  called  the 
perfect-gas  equation: 

p  v  =  N  R  T. 

The  numerical  value  of  the  gas-constant  R  depends  on  the  units 
in  which  the  pressure,  volume,  and  temperature  are  expressed.  In  sci- 
entific work,  temperature  is  always  expressed  in  centigrade  degrees 
(here  on  the  absolute  scale) ;  volume  is  ordinarily  expressed  in  cubic 
centimeters  or  liters;  and  pressure  in  dynes  per  square  centimeter, 
millimeters  of  mercury,  or  atmospheres.  One  dyne  is  a  force  of  such 
magnitude  that  when  it  acts  on  a  freely  moving  mass  of  one  gram  it 
increases  its  velocity  each  second  by  one  centimeter  per  second;  and 
the  pressure  of  one  dyne  per  square  centimeter  is  called  one  ~bar,  108 
bars  being  called  one  megdbar.^  One  atmosphere  is  a  pressure  equal 
to  that  exerted  by  a  column  of  mercury  76  cm.  in  height  at  0°  at  the 
sea-level  in  a  latitude  of  45°.  The  value  of  the  gas-constant  R  when 

*This  unit-quantity  is  called  by  some  authors  one  gram-molecular  weight 
or  one  gram-molecule  of  the  substance. 

tThe  megabar  is  a  unit  of  the  same  general  magnitude  as  the  atmosphere, 
and  should  replace  it  in  scientific  work ;  but  unfortunately  it  is  not  yet  generally 
employed. 


PRESSURE-VOLUME  RELATIONS  OF  GASES  9 

the  pressure  is  in  atmospheres  and  the  volume  is  in  cubic  centimeters 
is  82.07  (approximately  82). 

Prob.  3.  Calculate  the  value  of  the  atmosphere  in  dynes  per  sq.  cm. 
and  in  megabars.  The  density  of  mercury  at  0°  is  13.60.  The  force  of 
gravity  acting  on  any  freely  moving  body  increases  its  velocity  each  sec- 
ond by  g  centimeters  per  second.  The  value  of  g  at  the  sea-level  in  a 
latitude  of  45°  is  980.6  (approximately  9'80)  centimeters  per  second. 

Prob.  4.  Calculate  precisely  the  value  of  the  gas-constant  R  when 
the  pressure  is  in  atmospheres  and  the  volume  in  cubic  centimeters,  from 
the  fact  that  one  gram  of  oxygen  has  at  0°  and  0.1  atm.  a  volume  of 
7005  ccin.  Assume  in  this  problem  and  the  following  ones  that  the  sub- 
stance behaves  as  a  perfect  gas. 

Pro 6.  5.  Seven  grams  of  a  certain  gas  have  a  volume  of  6.35  1.  at 
20°  and  720  mm.  How  many  grams  make  one  mol? 

Pro 6.  6.  What  is  the  volume  occupied  by  12  g.  of  ether  vapor 
(C4H10O)  at  80°  and  600  mm.?  What  is  the  density?  What  is  the 
ratio  of  the  density  to  that  of  oxygen  at  the  same  temperature  and 
pressure? 

Pro&.  7.  How  many  grams  of  iron  must  be  taken  to  produce  by  its 
action  on  sulphuric  acid  one  liter  of  hydrogen  (H2)  at  27°  and  1  atm.? 

15.  Dalton's  Law  of  Partial  Pressures. — In  a  mixture  of  perfect 
gases  each  chemical  substance  (denned  as  in  Art.  5  to  be  a  substance 
consisting  of  a  particular  kind  of  molecule)  has  the  same  pressure  as 
it  would  if  it  were  alone  present  in  the  volume  occupied  by  the  mixture. 
The  pressures  of  the  separate  substances  are  called  partial  pressures. 

Partial  pressures  have  been  experimentally  determined  in  certain 
cases.  Thus,  when  a  platinum  vessel  containing  a  mixture  of  hydrogen 
and  nitrogen  at  a  high  temperature  is  immersed  in  an  atmosphere  of 
hydrogen,  hydrogen  passes  through  the  platinum  walls  until  its  pres- 
sures within  and  without  the  vessel  become  equal ;  the  difference  then 
observed  between  the  total  pressure  of  the  mixture  within  the  vessel 
and  the  pressure  of  the  hydrogen  outside  is  due  to  the  nitrogen,  which 
does  not  pass  through  the  wall ;  and  this  difference  is  equal  to  its  partial 
pressure.  Walls  of  this  kind,  permeable  for  only  one  of  the  substances 
present  in  a  mixture,  are  called  semipermeable  walls. 

The  principal  evidence  in  support  of  Dalton's  law  is,  however,  the 
fact  that  the  total  pressure  of  a  mixture  of  perfect  gases  is  actually 
equal  to  the  sum  of  the  partial  pressures  calculated  by  the  law. 

By  the  molal  composition  of  a  mixture  is  meant  its  composition 
expressed  in  terms  of  the  number  of  mols  of  each  of  the  substances. 


10  MOLAL  PROPERTIES  OF  GASES 

The  ratio  of  the  number  of  mols  of  any  one  substance  to  the  total 
number  of  mols  in  the  mixture  is  called  the  mol- fraction  x  of  that 
substance;  that  is,  a?±.=  N1/(N1  +  N2  +  .  .)•  The  partial  pressure 
of  each  substance  in  a  mixture  of  perfect  gases  is  evidently  equal  to  the 
product  of  its  mol-f  raction  by  the  total  pressure  of  the  mixture. 

Pure  dry  air  contains  21.0  mol-percent  of  oxygen  (O2),  78.1  mol-per- 
cent  of  nitrogen  (N2),  and  0.9  mol-percent  of  argon  (A).  The  corre- 
sponding value  of  one  mol  of  air  is  29.00  grams,  in  agreement  with  the 
value  directly  derived  from  density  measurements.  With  the  aid  of 
this  value  the  perfect-gas  equation  may  be  applied  directly  to  air. 

Prol.  8.  a.  From  the  composition  of  air  given  above,  calculate  the 
partial  pressures  of  the  separate  substances  in  millimeters  of  mercury 
when  the  barometer  stands  at  1  atm.  6.  Calculate  the  number  of  grams 
which  make  one  mol  of  air.  c.  Calculate  the  percent  by  weight  of  oxygen 
in  air. 

Prob.  9.  A  Bessemer  converter  is  charged  with  10,000  kilos  of 
iron  containing  3%  carbon.  How  many  cubic  meters  of  air  at  27° 
and  1  atm.  are  needed  for  the  combustion  of  all  the  carbon,  assuming 
one-third  to  burn  to  CO2  and  two-thirds  to  CO?  What  are  the  partial 
pressures  of  the  gases  evolved? 

Prob.  10.  At  50°  and  500  mm.,  nitrogen  tetroxide  has  a  density 
2.15  times  that  of  air  under  the  same  conditions.  What  percentage 
dissociation  according  to  the  equation  N2O4  =  2NO2  must  be  assumed 
to  explain  this  value?  What  is  the  partial  pressure  of  each  gas 
(NO,  and  N2O4)  in  the  mixture? 

16.  Determination  of  Molecular  Weights. — The  experimental  deter- 
mination of  the  molecular  weight  of  a  gaseous  substance  consists  in 
measuring  the  volume  of  a  weighed  quantity  of  the  substance  at  an 
observed  pressure  and  temperature.  From  these  quantities  (v,  m,  p, 
and  T)  the  molecular  weight  of  the  substance  is  calculated  with  the  aid 
of  the  perfect-gas  equation. 

Prob.  11.  Determination  of  Vapor-Density  by  Hofmann's  Method. — 
0.1035  g.  of  a  volatile  liquid  is  introduced  into  the  vacuum  above  a  mer- 
cury column  in  a  graduated  tube  standing  in  an  open  vessel  of  mercury. 
The  tube  is  entirely  surrounded  by  a  jacket  through  which  steam  at  100° 
is  passed.  The  mercury  column  falls  till  it  stands  260  mm.  above  the 
mercury-level  in  the  vessel  below,  and  the  volume  of  the  completely 
vaporized  liquid  is  observed  to  be  63.0  ccm.  The  barometric  pressure  at 
the  time  is  752  mm.  At  100°  the  density  of  mercury  is  13.35,  and  its  vapor- 
pressure  is  negligible.  Calculate  the  density  and  the  molecular  weight  of 
the  vapor. 


PRESSURE-VOLUME  RELATIONS  OF  GASES  11 

Since  gases  at  atmospheric  pressure  conform  to  the  perfect-gas 
laws  only  approximately,  and  since  gaseous  densities  are  not  commonly 
determined  with  so  great  accuracy  as  the  composition  of  substances, 
the  exact  value  of  the  molecular  weight  of  a  compound  is  usually  de- 
rived from  the  analytical  data,  the  density  being  employed  only  to 
determine  what  multiple  or  submultiple  of  the  value  so  derived  is  in 
accordance  with  Avogadro's  principle. 

Prol.  12.  a.  A  certain  oxide  contains  exactly  72.73%  of  oxygen. 
What  does  this  show  in  regard  to  its  molecular  weight?  6.  At  0°  and 
1  atm.,  one  liter  of  this  (gaseous)  oxide  is  found  to  weigh  exactly  1.977  g. 
What  is  the  molecular  weight  of  the  oxide  corresponding  to  this  datum? 
c.  What  is  the  exact  molecular  weight  derived  by  considering  the  data 
relating  both  to  the  composition  and  density?  d.  What  do  these  two  values 
of  the  molecular  weight  show  as  to  the  percentage  deviation  of  the  density 
from  that  required  by  the  perfect-gas  equation? 

17.  Derivation  of  the  Atomic  Weights  of  Elements  and  of  the 
Molecular  Composition  of  Compounds. — The  exact  values  of  atomic 
weights  are  based  on  analytical  determinations  of  the  combining 
weights,  as  described  in  Art.  4.  The  multiple  of  the  combining  weight 
adopted  as  the  atomic  weight  of  any  element  is  derived  by  finding  the 
smallest  weight  of  the  element  contained  in  one  molecular  weight  of 
any  of  its  gaseous  compounds.  This  is  the  true  atomic  weight  only  in 
case  some  one  of  the  compounds  studied  contains  in  its  molecule  a 
single  atom  of  the  element ;  and  the  adopted  atomic  weight  is  therefore 
strictly  only  a  maximum  value,  of  which  the  true  atomic  weight  may 
be  a  submultiple.  The  probability  that  the  true  atomic  weight  has  been 
found  evidently  increases  with  the  number  of  the  gaseous  compounds 
whose  molecular  weights  have  been  determined. 

The  multiples  of  the  combining  weights  adopted  as  the  atomic- 
weights  have,  however,  not  been  derived  solely  from  molecular-weight 
determinations.  From  the  laws  relating  to  certain  other  properties, 
such  as  the  heat-capacities  of  gaseous  and  solid  substances  (considered 
in  Arts.  118  and  120),  independent  values  of  the  atomic  weights  have 
been  obtained,  which  confirm  and  extend  those  derived  from  molecular 
weights. 

The  molecular  composition  of  a  gaseous  substance  (that  is,  the 
number  and  nature  of  the  atoms  in  its  molecule)  can  evidently  be 
derived  from  its  molecular  weight,  its  composition  by  weight,  and  the 
atomic  weights  of  the  elements  contained  in  it.  The  chemical  formulas 


12  HOLAL  PROPERTIES  OF  GASES 

of  substances  whose  molecular  weights  in  the  gaseous  state  are  known 
are  ordinarily  so  written  as  to  express  this  molecular  composition. 
Such  formulas  are  called  molecular  formulas. 

Prob.  13.  Certain  oxygen  compounds  have  molecular  weights  M 
(referred  to  that  of  oxygen  as  32)  and  percentages  of  oxygen  x  as  fol- 
lows :  Sulphur  trioxide,  M  =  80.07,  x  —  59.95 ;  water,  M  =  18.02,  x  = 
88.80 ;  carbon  dioxide,  M  —  44.00,  x  =  72.73.  a.  Find  the  weight  of 
oxygen  contained  in  one  molecular  weight  of  each  of  these  oxides. 
&.  State  how  these  values  show  that  the  atomic  weight  of  the  element 
oxygen  is  one-half  of  the  assumed  molecular  weight  of  oxygen  gas,  and 
therefore  that  the  molecule  of  oxygen  gas  consists  of  two  atoms. 

Prob.  Hi.  a.  Calculate  the  combining  weight  of  the  element  contained 
in  the  oxide  whose  composition  was  given  in  Prob.  12a.  &.  What  conclu- 
sion as  to  its  atomic  weight  can  be  drawn  from  this  combining  weight 
and  the  density  given  in  Prob.  12  &?  c.  What  conclusion  can  be  drawn 
as  to  the  molecular  formula  of  the  oxide? 

Pro 6.  15.  a.  The  chloride  of  a  certain  element  is  found  by  analysis 
to  contain  52.50%  Cl,  whose  atomic  weight  is  35.46;  and  it  is  found  to 
have  at  150°  a  vapor-density  4.71  as  great  as  that  of  air.  What  conclusion 
can  be  drawn  from  these  facts  as  to  the  exact  atomic  weight  of  the  ele- 
ment? &.  The  hydride  of  the  same  element  is  a  gas  which  contains  5.91% 
of  hydrogen ;  and  it  is  found  to  be  produced  without  change  in  volume 
when  hydrogen  (H2),  whose  atomic  weight  is  1.008,  is  passed  over  the 
solid  elementary  substance.  What  conclusion  can  now  be  drawn  as  to  the 
atomic  weight  of  the  element?  c.  What  are  the  simplest  molecular 
formulas  of  the  hydride  and  of  the  chloride  consistent  with  these  con- 
clusions? 

Pro6.  16.  A  certain  hydrocarbon  is  composed  of  92.25%  of  carbon 
and  7.75%  of  hydrogen,  whose  atomic  weights  are  12.00  and  1.008. 
respectively.  Its  density  in  the  form  of  vapor  at  100°  and  1  atmosphere 
is  2.47  times  as  great  as  that  of  oxygen  under  the  same  conditions. 
Calculate  its  exact  molecular  weight,  and  derive  its  molecular  formula. 

The  molecules  of  elementary  substances  may  consist  of  single 
atoms  or  of  two  or  more  atoms.  Thus  the  molecular  formulas  of 
some  of  those  whose  density  in  the  gaseous  state  has  been  determined 
are:  H2,  N2,  02,  F2,  G12,  Br2,  I2,  P4,  As4,  He,  A,  Hg,  Cd,  Zn.  At  very 
high  temperatures  some  of  these  have  been  shown  to  dissociate  into 
simpler  molecules;  thus  above  1500°  the  molecule  of  iodine  consists 
of  a  single  atom. 

A  knowledge  of  the  molecular  formulas  of  substances  is  of  impor- 
tance principally  because  the  chemical  relationships  of  different  sub- 
stances are  far  more  clearly  brought  out  by  such  formulas  than  by 
the  simpler  ones  expressing  merely  composition  by  weight.  The 


PRESSURE-VOLUME  RELATIONS  OF  GASES 


13 


structure  theory,  which  underlies  the  science  of  organic  chemistry, 
is  based  upon  a  knowledge  of  the  molecular  weights  of  substances. 

18.  Deviations  from  the  Perfect-Gas  Laws  at  Moderate  Pressures. 
Prob.  17.    Calculate  accurately  the  percentage  deviations  at  0°  and 

1  atin.  of  the  volumes  of  one  mol  of  oxygen  and  of  one  mol  of  nitrous 
oxide  from  that  of  a  perfect  gasj  At  0°  and  1  atin.  the  density  of 
oxygen  is  0.001429,  and  that  of  nitrous  oxide  (N2O)  is  0.001978.x 

The  deviations  from  the  perfect-gas  law,  so  long  as  they  do  not  ex- 
ceed a  few  percent,  are  accurately  expressed  by  the  equation  p  v  = 
N  E '  T  (1  -f-  cxp),  in  which  ex  is  an  empirical  coefficient  which  can  be 
determined  for  each  gas  at  each  temperature  from  a  measurement  of 
p  v  at  some  one  pressure.  The  following  table  shows  the  values  of  100  <*, 
representing  the  percentage  deviation  of  p  v  from  N  R  T  at  one  atmos- 
phere, for  various  gases  at  0°,  together  with  their  condensation-tem- 
peratures. 
Formula  of  gas.  He  H2  N2  NO  CO2  NH8  SO, 

100  « _j_0.06    +0.05    —0.04    —0.12    —0.68   —1.52  —2.38 

Condensation-temp.    —269°    —253°   —196°    —151°    —78°    —34°    — 103 

19.  Pressure- Volume  Relations  of  Gases  at  High  Pressures.— In 
Figure  1  are  plotted  the  values  of  p  v/T  (in  arbitrary  units)  as  ordi- 


16 


V 


V 


40         80        I2O       l6o       2OO       240       28O       33O 

Pressure 

PIGDBB  i 


14  MOLAL  PROPERTIES  OF  GASES 

nates  against  the  values  of  the  pressure  (in  meters  of  mercury)  as 
abscissas  for  one  mol  of  hydrogen  at  60°,  of  nitrogen  at  60°,  of  carbon 
dioxide  at  60°  and  at  40°,  and  of  a  perfect  gas  (marked  P.  G.  in  the 
figure).  At  temperatures  between  0°  and  60°,  helium  (He)  and  neon 
(Ne)  have  curves  similar  to  that  of  hydrogen  (Hj) ;  oxygen  (O2), 
carbon  monoxide  (CO),  and  nitric  oxide  (NO)  have  curves  similar 
to  that  of  nitrogen;  and  nitrous  oxide  (N2O),  ammonia  (NH8),  and 
ethylene  (C2H4)  have  curves  similar  to  those  of  carbon  dioxide  (COa). 

ProT).  18.  Summarize  the  general  conclusions  in  regard  to  the 
pressure-volume  relations  of  gases  that  can  be  drawn  from  the  curves 
of  Figure  1,  the  statements  in  the  preceding  text,  and  the  condensation- 
temperatures  given  in  Art.  18. 

Pro6.  19.  Estimate  with  the  aid  of  the  figure  the  ratio  of  the  volume 
of  one  mol  of  each  of  the  gases  at  40  m.  (or  53  atm.)  to  that  which  one 
mol  of  a  perfect  gas  would  have  at  the  same  temperature  and  pressure. 


CHAPTER   III 

PROPERTIES  OF  SOLUTIONS  RELATED  TO 
MOLECULAR  COMPOSITION 


VAPORY-PRESSURE    AND    BOILING-POINT    IN    GENERAL 

20.  Vapor-Pressure. — A  liquid  in  contact  with  a  vacuous  space 
vaporizes  until  the  pressure  of  its  vapor  in  that  space  attains  a 
perfectly  definite  value  which  is  determined  by  the  nature  of  the 
liquid  and  by  the  temperature.  If,  on  the  other  hand,  vapor  having 
a  pressure  greater  than  this  definite  value  is  brought  into  contact 
with  the  liquid,  condensation  occurs  until  the  pressure  of  the  vapor 
falls  to  that  value.  In  other  words,  for  a  given  liquid  at  a  given 
temperature  there  is  only  one  pressure  which  its  vapor  can  have  and 
exist  in  equilibrium  with  that  liquid.  This  pressure  is  called  the 
vapor-pressure  of  the  liquid.  This  is  to  be  distinguished  from  the 
pressure  of  the  vapor,  which  when  not  in  contact  with  the  liquid 
may  have  any  value  from  zero  up  to  one  somewhat  exceeding  the 
vapor-pressure.  Solids  likewise  have  definite  vapor-pressures,  which 
with  certain  substances  (like  iodine)  are  appreciable  even  at  room 
temperature. 

The  vapor-pressure  of  a  liquid  or  solid  substance  increases  rapidly 
with  increasing  temperature,  as  illustrated  by  the  data  of  Prob.  3  below. 

When  a  liquid  is  in  contact  with  a  space  containing  a  gas  (for 
example,  when  water  is  in  contact  with  an  air  space),  approximately 
the  same  quantity  of  the  liquid  vaporizes  as  if  the  gas  were  not 
present,  provided  the  gas  is  only  slightly  soluble  in  the  liquid,  and 
provided  its  pressure  is  not  much  greater  than  one  atmosphere.  When 
the  gas  is  readily  soluble  in  the  liquid,  or  when  its  pressure  is  large, 
considerable  deviations  from  this  principle  may  result. 

Pro6.  1.  At  28°  and  1  atm.  25  ccm.  of  dry  air  are  collected  over 
water,  whose  vapor-pressure  at  28°  is  28  mm.  a.  What  is  the  pressure 
if  the  volume  is  still  25  ccm.?  ft.  What  is  the  volume  if  the  pressure  is 
still  1  atm.? 

Pro6.  2. — Mr-Bubbling  Method  of  Determining  Vapor-Pressure. — 
2000  ccm.  of  dry  air  at  15°  and  760  mm.  are  bubbled  through  bulbs  con- 
taining a  known  weight  of  carbon  bisulphide  (CS2)  at  15°,  and  the  mix- 

15 


16  MOLAL  PROPERTIES  OF  SOLUTIONS 

ture  of  air  and  bisulphide  vapor  is  allowed  to  escape  into  the  air  at  a 
pressure  of  760  mm.  By  reweighing  the  bulbs,  3.011  g.  of  the  bisulphide 
are  found  to  have  vaporized.  What  is  the  vapor-pressure  of  carbon  bisul- 
phide at  15°  ? 

Steam-Distillation  of  Liquids  Insoluble  in  Water. — 

Prob.  3.  Steam  is  bubbled  through  chlorbenzene  (C6H8C1)  in  a  distill- 
ing flask ;  and  the  vapors,  which  escape  under  a  barometric  pressure  of 
1  atm.,  are  condensed  as  a  distillate.  The  steam  partially  condenses  in 
the  distilling  flask,  and  brings  the  mixture  of  water  and  chlorbenzene 
(which  are  not  appreciably  soluble  in  one  another)  to  that  temperature 
where  equilibrium  prevails  between  each  liquid  and  its  vapor.  Determine 
this  temperature  and  the  molal  composition  of  the  distillate  with  the 
aid  of  a  plot  of  the  following  data,  which  represent  the  vapor-pressures 
of  the  pure  substances  at  various  temperatures : 

70°  80°  90°          100° 

Water  ....        234  355  526  760  mm. 

Chlorbenzene        ...          98  145  208  292  mm. 

Prob.  4-  A  current  of  steam  is  passed  at  atmospheric  pressure  through 
a  mixture  of  water  and  nitrobenzene  (C6H6NO2).  Calculate  the  tempera- 
ture of  the  distilling  mixture  and  the  percentage  by  weight  of  nitroben- 
zene in  the  distillate  from  the  following  data:  the  vapor-pressure  of 
water  at  100°  is  760  mm.  and  changes  by  3.58%  per  degree ;  that  of  nitro- 
benzene at  100°  is  20.9  mm.  and  changes  by  5.0%  per  degree. 

21.  Relation  of  Boiling-Point  to  Vapor-Pressure.— The  'boiling- 
point  of  a  liquid  is  the  temperature  at  which  it  is  in  equilibrium  with 
its  vapor  when  both  are  subjected  to  any  definite  external  pressure. 
In  other  words,  it  is  the  temperature  at  which  the  vapor-pressure,  which 
increases  as  the  temperature  rises,  becomes  equal  to  the  external  pres- 
sure.  When  this  temperature  is  exceeded  by  an  infinitesimal  amount, 
assuming  that  there  is  no  superheating,  the  vapor  forms  throughout  the 
mass  of  the  liquid  (not  merely  at  its  free  surface),  giving  rise  to  the 
familiar  phenomenon  of  boiling. 

Prob.  5.  The  vapor-pressure  of  water  at  100°  increases  27.2  mm.  per 
degree.  What  variation  of  its  boiling-point  corresponds  to  a  variation  of 
the  barometric  pressure  from  730  to  790  mm.  ? 

22.  Change  of  Vapor-Pressure  with  Temperature.   The  Clapeyron 
Equation. — From  the  laws  of  thermodynamics  it  can  be  shown  that  the 
rate  at  which  the  vapor-pressure  p  of  a  liquid  or  solid  at  the  absolute 
temperature  T  increases  with  the  temperature  is  expressed  exactly  by 
the  Clapeyron  equation : 

dp  AH 

dT  =  =  (v-v0)T 


VAPOR-PRESSURE  AND  BOILING-POINT  17 

In  this  equation  v  represents  the  volume  of  one  mol  of  the  (.saturated) 
vapor  at  the  pressure  p  and  temperature  T,  v0  is  the  volume  of  one 
mol  of  the  liquid  or  solid  substance  under  the  same  conditions,  and 
A  H  is  the  increase  in  the  heat-content  of  the  substance,  equal  to  the 
heat  withdrawn  from  the  surroundings,  when  one  mol  of  it  vaporizes 
at  the  temperature  T.  The  quantity  Ajff  is  called  the  molal  heat  of 
vaporization. 

Since  at  the  boiling-point  of  a  liquid  its  vapor-pressure  is  equal 
to  the  external  pressure  upon  the  liquid  and  vapor,  the  Clapeyron 
equation  also  expresses  (more  clearly  in  the  inverted  form)  the  change 
of  boiling-point  with  the  external  pressure. 

In  numerical  applications  of  this  equation,  the  energy  quantities 
&H  and  (v  --  v^)dp  must  be  expressed  in  corresponding  units.  The 
latter  quantity  will  be  in  ergs  when  the  volumes  are  in  cubic  centi- 
meters and  the  pressure  is  in  dynes  per  square  centimeter. 

Three  units  of  energy  are  commonly  used  in  scientific  work — the 
erg,  the  joule,  and  the  calorie.  The  erg  is  the  work  done  when  a 
force  of  one  dyne  is  displaced  through  one  centimeter.  The  joule  is 
a  decimal  multiple  of  the  erg;  namely,  one  joule  equals  107  ergs.  The 
mean  calorie  is  one  one-hundredth  part  of  the  heat  required  to  raise 
one  gram  of  water  from  0°  to  100°.  This  is  identical  within  0.02% 
with  the  ordinary  calorie,  which  is  the  heat  required  to  raise  one  gram 
of  water  from  15°  to  16°.  One  calorie  (1  cal.)  is  equal  to  4.182  (approx- 
imately 4.18)  X  1°7  ergs>  this  value  being  the  so-called  mechanical 
equivalent  of  heat. 

Pro  6.  6.  a.  Calculate  w'ith  the  aid  of  the  Clapeyron  equation  the 
volume  of  one  mol  of  saturated  water-vapor  at  100°  from  the  following 
data:  At  100°  the  vapor-pressure  of  water  increases  27.2  mm.  per  de- 
gree, the  heat  of  vaporization  of  one  gram  of  it  is  537  cal.,  and  the 
specific  volume  (i.e.,  the  volume  of  one  gram)  of  liquid  water  is  1.043. 
&.  Calculate  by  th$  perfect-gas  equation  the  volume  of  one  mol  of  satu- 
rated water-vapor  at  100°.  c.  By  comparing  the  two  values  of  this 
quantity  obtained  in  a  and  &,  determine  the  percentage  error  made  In 
assuming  that  the  saturated  vapor  conforms  to  the  perfect-gas  law. 

Note. — The  molal  volume  of  the  vapor  of  alcohol  saturated  at  78.3° 
(where  the  pressure  is  760  mm.)  is  3.6%  smaller,  and  that  of  the  vapor 
of  ether  saturated  at  12.9°  (where  the  pressure  is  330  mm.)  is  3.5% 
smaller,  than  the  molal  volume  calculated  by  the  perfect-gas  equation. 

Pro 6.  7.    a.  Derive  from  the  Clapeyron  equation  the  simpler,  but 

less  exact,  expression  -   -=-2-  —  by  assuming  that  the  volume  of 


18  MOLAL  PROPERTIES  OF  SOLUTIONS 

the  liquid  is  negligible  in  comparison  with  that  of  the  vapor  and  that  the 
vapor  conforms  to  the  perfect-gas  laws.  I).  What  percentage  error  in 
dp /d,T  would  result  from  each  of  these  assumptions  if  it  be  calculated 
for  water  at  100°  by  this  approximate  equation  with  the  aid  of  the 
data  of  Prob.  Gal  c.  In  numerical  applications  of  this  equation  R 
and  &H  must  be  expressed  in  corresponding  units.  Show  from  the  data 
of  Prob.  4,  Art.  14,  that  the  value  of  R  is  8.316  (approximately  8.32)  in 
joules  per  degree,  and  1.9885  (approximately  1.99)  in  calories  per  degree. 
Pro&.  8.  a.  Integrate  the  approximate  Clapeyron  equation  so  as  to 
obtain  a  relation  between  the  vapor-pressures  pt  and  p2  at  two  different 
temperatures  T,  and  Tv  Assume  that  the  heat  of  vaporization  does 
not  vary  between  the  two  temperatures.  &.  Calculate  by  the  equation 
so  obtained  the  boiling-point  of  water  at  92  mm.,  and  its  vapor-pressure 
at  75°.  Compare  these  calculated  values  with  the  actual  ones  given 
above,  c.  State  what  inexact  assumptions  are  involved  in  the  equation 
which  would  account  for  the  divergence. 

23.  Solutions. — A  solution  is  a  physically  homogeneous  mixture  of 
two  or  more  chemical  substances;  that  is,  one  which  has  no  larger 
aggregates  than  the  molecules  themselves.  Solutions  thus  defined  may 
be  gaseous,  liquid,  or  solid ;  but  only  liquid  solutions  will  be  here  con- 
sidered. When  one  substance  is  present  in  large  proportion  it  is  called 
the  solvent,  and  any  substance  present  in  small  proportion  is  called  a 
solute. 

The  composition  of  solutions  is  often  expressed  in  terms  of  the 
mol-fractions  of  the  substances,  defined  as  in  Art.  15. 

In  considering  the  equilibrium  of  solutions  with  the  vapor  or  with 
the  solid  solvent,  the  term  phase  is  conveniently  employed.  The  phases 
of  a  system  are  its  physically  homogeneous  parts,  separated  from  one 
another  by  physical  boundaries.  Thus  any  gaseous  mixture  or  any 
solution  or  any  solid  substance  f orms  a  single  phase.  A  system  may 
consist  of  any  number  of  such  phases.  Thus  a  solution  in  contact 
with  its  vapor,  or  with  the  solid  solvent,  or  with  the  solid  solute,  is 
an  example  of  a  two-phase  system.  A  .solution  in  contact  both  with 
the  vapor  and  the  solid  solvent  is  a  three-phase  system. 

With  reference  to  the  proportions  in  which  the  substances  are 
present,  two  groups  of  solutions  may  be  distinguished :  dilute  solutions, 
those  in  which  the  mol-fraction  of  the  solute  is  small  (not  greater  than 
0.01  or  0.02) ;  and  concentrated  solutions,  those  in  which  each  sub- 
stance is  present  in  considerable  proportion.  There  is,  of  course,  no 
sharp  line  of  demarcation  between  these  two  groups  of  solutions. 


VAPOR-PRESSURE  AXD  BOILING-POIXT  11) 

Some  types  of  concentrated  solutions  and  all  dilute  solutions  con- 
form approximately — more  closely  as  the  mol-fraction  of  the  solute 
approaches  zero — to  certain  laws,  which,  in  analogy  with  the  laws  of 
perfect  gases,  may  be  called  the  laws  of  perfect  solutions.  The  funda- 
mental laws  of  perfect  solutions  are  the  vapor-pressure  laws  of  Raoult 
and  Henry,  and  the  corresponding  laws  of  distribution  between  phases 
and  of  the  osmotic  pressure  of  solutions.  To  consideration  of  these 
laws  the  rest  of  this  chapter  is  mainly  devoted. 

VAPOR-PRESSURE    AND    BOILING-POINT    OF    DILUTE    SOLUTIONS 

24.  Raoult 's  Law  of  Vapor-Pressure  Lowering. — The  addition 
of  a  solute  to  a  solvent  causes  at  any  temperature  a  fractional  lower- 
ing of  the  vapor-pressure  of  the  solvent  equal  to  the  mol-fraction  (#) 
of  the  solute.  That  is : 

Po  —  p  _  N 

Po         r  No  +  N  " 

where  p0  is  the  vapor-pressure  of  the  pure  solvenvand  p  is  its  vapor- 
pressure  above  a  solution  consisting  of  N  mols  of  solute  and  N0  mols 
of  solvent.*  In  this  expression  N0  is  equal  to  the  weight  m0  of  the 
solvent  divided  by  its  molecular  weight  M^  in  the  vapor,  and  N  is 
equal  to  the  weight  m  of  the  solute  divided  by  its  molecular  weight 
M  in  the  solution.  The  value  of  M0  is  ordinarily  that  corresponding 
to  the  molecular  formula  of  the  solvent;  for,  as  stated  in  Art.  9,  the 
molecular  formula  is  commonly  so  written  as  to  represent  the  molecular 
weight  of  the  substance  in  the  state  of  a  perfect  gas. 

In  the  case  of  very  dilute  solutions  the  mol-fraction  N/(N0  +  N) 
may  evidently  be  replaced  by  the  mol-ratio  N /N»  without  causing 
appreciable  error. 

When  the  solute  is  so  volatile  as  to  have  an  appreciable  vapor 
pressure,  the  quantity  p  in  the  Raoult  equation  denotes,  of  course, 
the  partial  vapor-pressure  of  the  solvent,  not  the  total  vapor-pressure 
of  the  solution. 

Although  Raoult's  law  is  exact  only  in  the  case  of  very  dilute 
solutions,  it  holds  true  approximately  up  to  moderate  concentrations 
in  all  cases,  and  up  'to  very  high  concentrations  in  some  cases,  as 
will  be  described  in  Art.  30. 

*Throughout  this  chapter  quantities  referring  to  the  solvent  are  represented 
by  letters  with  the  subscript  zero,  those  referring  to  the  solution  or  solute  by 
letters  without  subscripts. 


20  MOLAL    PROPERTIES  OF  SOLUTIONS 

Determination  of  Molecular  Weights  and  Molecular  Composition. — 
Pro&.  9.    At  30°  the  vapor-pressure  of  ethyl  alcohol    (C2H5OH)    is 
78.0  mm.,  and  that  of  an  alcohol   solution  containing  5%   of  a  non- 
volatile substance  is  75.0  mm.     What  is  the  molecular  weight  of  the 
substance? 

Pro ~b.  10.  The  experiment  described  in  Prob.  2,  Art.  20,  was  repeated, 
using  in  place  of  pure  carbon  bisulphide  an  8.00%  solution  of  sulphur  in 
carbon  bisulphide.  2.902  g.  of  carbon  bisulphide  were  found  to  have 
vaporized.  Calculate  the  molecular  weight  of  the  sulphur,  and  find  its 
molecular  formula. 

Raoult's  law  may  also  be  stated  in  the  following  simple  form,  which 
indicates  more  clearly  its  real  significance:  the  vapor-pressure  (p)  of 
the  solvent  in  a  perfect  solution  is  proportional  to  its  mol-fraction 
O0)  ;  that  is,  representing  by  p0  the  vapor-pressure  of  the  pure  solvent, 

P  =  Po  a?0. 

ProT).  11.  a.  Show  that  the  two  statements  of  Raoult's  law  are  mathe- 
matically equivalent.  6.  Show  that  the  second  statement  of  the  law 
requires  that  the  proportionality-factor  be  the  vapor-pressure  of  the  pure 
solvent,  as  is  assumed  in  the  mathematical  expression  of  it. 

Raoult's  law  relates  fundamentally  to  the  distribution  between  the 
liquid  phase  and  vapor  phase  of  the  chemical  substance  (Art.  5)  whose 
partial  pressure  in  the  vapor  is  under  consideration.  In  other  words, 
from  the  molecular  standpoint,  it  relates  to  the  distribution  of  the  kind 
of  molecules  which  give  rise  to  this  partial  pressure.  It  shows  that 
the  number  of  these  molecules  which  are  present  in  unit-volume  of  the 
Vapor  when  equilibrium  has  been  reached  is  proportional  to  the  ratio  in 
the  liquid  of  the  number  of  this  kind  of  molecule  to  the  total  number 
of  molecules  of  all  kinds.  Moreover,  since  this  molecule-ratio  is  unity 
for  a  solvent  which  consists  solely  of  the  kind  of  molecule  under  con- 
sideration, the  proportionality-factor  in  the  expression  of  the  law  is  the 
vapor-pressure  of/ the  pure  solvent. 

Raoult's  law  in  the  forms  considered  above  presupposes  that  the 
vapor  conforms  to  the  perfect-gas  law.  It  can  be  shown  that  the  devia- 
tion of  the  vapor  from  this  law,  when  expressed  (as  in  Art.  18)  by  the 
equation  p  v  =  R  T  (1  +  ap),  can  be  substantially  corrected  for,  so 
long  as  neither  the  mol-fraction  of  the  solute  nor  the  correction  term 
<x  p0  exceeds  5  percent,  by  writing  the  Raoult  equation  in  the  form : 
p(l  +  oc  p)=x0p0(l+  cxpj. 


AND  BOILING-POINT  21 

25.  Relation  of  Boiling-Point  Raising  to  Vapor-Pressure  Lower- 
ing for  Non-  Volatile  Solutes. 

Prob.  12.  a.  What  is  the  vapor-pressure  in  mm.  of  a  solution  at 
100°  containing  5  g.  of  glucose  (C6H12O6)  in  100  g.  of  water?  ft.  What 
is  its  boiling-point?  Its  vapor-pressure  at  100°,  like  that  of  water, 
increases  3.58%  per  degree. 

Prob.  13.  The  vapor-pressure  of  ethyl  alcohol  is  721.5  mm.  at  77°, 
751.0  at  78°,  781.5  at  79°,  and  813.0  at  80°.  a.  Plot  on  a  large  scale 
these  vapor-pressures  as  ordiuates  and  the  temperatures  as  abscissas. 
Calculate  the  vapor-pressures  at  78,  79,  and  80°  of  a  solution  consist- 
ing of  2  mols  of  a  solute  and  98  mols  of  alcohol,  and  of  one  consisting 
of  4  mols  of  solute  and  96  mols  of  alcohol  ;  and  plot  these  values  on 
the  diagram.  &.  With  the  aid  of  the  diagram  find  the  boiling-points 
at  1  atm.  of  pure  alcohol  and  of  the  two  solutions,  c.  Show  from  the 
geometrical  relations  of  the  diagram  that  the  raising  of  the  boiling- 
point  is  proportional  to  the  lowering  of  the  vapor-pressure  at  the 
boiling-point  of  the  solvent,  for  dilute  solutions  (for  which  the  graphs 
may  be  considered  to  be  parallel  straight  lines),  d.  Find  the  value  for 
ethyl  alcohol  of  the  proportionality-constant  involved  in  the  relation  just 
stated. 

Kepresenting  by  T  —  T0  the  raising  of  the  boiling-point  and  by 
Po  —  P  the  lowering  of  the  vapor-pressure  produced  by  increasing  the 
mol-fraction  of  the  solute  from  0  to  x,  and  representing  by  dT0/dp0  the 
reciprocal  of  the  rate  of  change  of  the  vapor-pressure  of  the  solvent 
with  the  temperature,  the  relations  derived  in  the  preceding  problem 
for  dilute  solutions  may  be  expressed  by  the  equation  : 


26.  Relation  between  Boiling-Point  Raising  and  Molal  Compo- 
sition. —  By  combining  the  expression  derived  in  the  preceding  article 
with  Raoult's  equation  for  vapor-pressure  lowering  there  is  obtained 
the  following  relation  between  the  raising  of  the  boiling-point  (T  —  T0) 
and  the  mol-fraction  x  of  the  solute  : 


The  quantity  — ±JL_  is  evidently  a  constant  characteristic  of  the  sol- 

dpo/po 

vent,  which  may  be  called   its   boiling-point  constant.    Another   ex- 
pression for  it,  in  terms  of  the  molal  heat  of  vaporization  of  the  solvent 


22  MOLAL  PROPERTIES  OF  SOLUTIONS 


,  can  be  derived  from  the  Clapeyron  equation.  Thus  from  the 
approximate  form  of  that  equation  is  obtained  at  once  the  following 
relation  : 

dT0 


Representing  the  boiling-point  constant  by  a  single  letter  B,  and 
noting  that,  so  long  as  the  mol-f  raction  of  the  solute  is  small,  it  is  sub- 
stantially equal  to  the  mol-ratio  N/N0,  the  law  of  boiling-point  raising 
for  dilute  solutions  may  be  expressed  by  the  equation  : 

N 
T  —  T  —  7? 

-to  —  &  T^~* 

No 

It  is  evident  from  this  equation  that  the  boiling-point  constant  B  is 
the  ratio  of  the  boiling-point  raising  to  the  number  of  mols  of  solute 
which  are  associated  with  one  mol  of  solvent  ;  or,  briefly,  it  is  the  boil- 
ing-point raising  per  mol  of  solute  in  one  mol  of  solvent.  In  chemical 
literature  is  commonly  recorded,  not  this  boiling-point  constant,  but 
another  constant,  called  the  molal  boiling-point  raising,  which  is  the 
boiling-point  raising  per  mol  of  solute  in  1000  grams  of  solvent. 

From  the  value  of  either  of  these  constants  for  a  given  solvent  may 
be  calculated  by  direct  proportion  the  actual  raising  of  the  boiling- 
point  caused  by  a  known  number  of  mols  of  any  solute  ;  or,  conversely, 
there  may  be  calculated  the  number  of  mols  of  solute  corresponding  to 
any  observed  raising  of  the  boiling-point.  Therefore  the  law  of  boiling- 
point  raising,  like  Raoult's  law  from  which  it  has  been  derived,  makes 
it  possible  to  determine  the  molecular  weight  of  substances  in  solution. 

The  values  of  the  constants  obtained  for  some  important  solvents  by 
the  three  methods  indicated  above  and  illustrated  by  Prob.  15  are  as 

°    OWS'  Water  Ethyl  ether    Ethyl  alcohol       Benzene 

HtO  CiH100  CzHjOH  CnHfi 


Boiling-point  constant  ........      28.6  28.5  25,8  34.0 

Molal  boiling-point  raising  ----       0.515  2.11  1.19  2.65 

Prob.  l!f.  a.  Calculate  the  molal  boiling-point  raising  for  water  from 
its  boiling-point  constant.  &.  Formulate  the  algebraic  relation  between 
the  two  constants. 

Prob.  15.  —  Methods  of  Determining  the  Boiling-Point  Constant.  — 
Calculate  the  boiling-point  constant  for  ethyl  alcohol  from  the  follow- 
ing data.  a.  The  heat  of  vaporization  of  one  gram  is  206  cal.  at  the 
boiling-point  78.3°.  6.  Its  vapor-pressure  has  the  values  given  in 
Prob.  13.  c.  The  boiling-point  of  a  solution  of  1  g.  naphthalene  (C10H8) 
in  50  g.  alcohol  is  0.185°  higher  than  that  of  pure  alcohol. 


VAPOR-PRESSURE  AND  BOILING-POINT  23 

Determination  of  Molecular  Weights  and  Molecular  Composition. — 
P/-O&.  16.    a.    When  10.6  g.  of  a  substance  are  dissolved  in  740  g.  of 
ether  (C4H10O),  its  boiling-point  is  raised  0.284°.    What  is  the  molecular 
weight  of  the  substance?    6.  The  substance  is  a  hydrocarbon  containing 
90.50%  carbon.    What  is  its  molecular  formula? 

Profc.  17.  A  solution  of  3.04  g.  benzoic  acid  in  100  g.  ethyl  alcohol 
boils  0.288°  higher  than  pure  alcohol.  A  solution  of  6.34  g.  benzoic  acid 
in  100  g.  benzene  boils  0.696°  higher  than  pure  benzene.  Calculate  the 
molecular  weight  of  benzoic  acid  in  each  of  these  solvents,  and  state 
what  the  results  show  in  regard  to  its  molecular  formula  in  each  sol- 
vent. Its  composition  by  weight  is  expressed  by  the  formula  C6H6CO2H. 

The  molecular  weights  of  substances  are  ordinarily  found  to  be  the 
same  in  the  dissolved  state  as  in  the  gaseous  state ;  but  hydroxyl  com- 
pounds (such  as  the  alcohols  and  organic  acids)  form  in  non-oxygen- 
ated solvents  (such  as  benzene  or  chloroform)  double  or  even  more 
highly  associated  molecules.  This  indicates  that  the  molecules  of  hy- 
droxyl compounds  are  associated  also  in  the  state  of  pure  liquids. 
Oxygenated  solvents  (such  as  water,  alcohols,  acetic  acid,  ether,  and 
acetone)  have  the  power  of  breaking  down  these  associated  molecules 
into  the  simple  ones. 

27.    Partial  Vapor-Pressure  of  Volatile  Solutes.     Henry's  Law. 

In  addition  to  Eaoult's  law,  which  relates  to  the  vapor-pressure  of  the 
solvent  in  a  solution  containing  a  small  proportion  of  a  solute,  there  is 
another  fundamental  law  which  relates  to  the  vapor-pressure  of  the 
solute  in  such  a  solution.  This  law,  known  as  Henry's  Law,  may  be 
stated  as  follows.  The  partial  vapor-pressure  of  any  chemical  substance 
present  in  small  proportion  in  a  solution  is  proportional  to  its  mol- 
fraction.  That  is, 

p  =  kx, 

where  Tc  is  a  proportionality-constant  dependent  on  the  nature  of  the 
substance  and  of  the  solvent  and  on  the  temperature. 

Prob.  18.  The  total  vapor-pressure  of  a  solution  containing  3%  by 
weight  of  ethyl  alcohol  in  water  is  760  mm.  at  97.11°,  and  the  vapor- 
pressure  of  pure  water  at  this  temperature  is  685  mm.  Calculate  with  the 
help  of  Raoult's  law  and  of  Henry's  law  the  partial  pressures  at  97.11° 
of  ethyl  alcohol  and  water  in  a  solution  containing  2.00  mol-percent  of 
ethyl  alcohol. 

It  is  often  convenient  in  the  case  of  dilute  solutions  to  express  the 
composition  in  terms  of  concentration,  instead  of  mol-fraction.  By  the 


24  MOLAL  PROPERTIES  OF  SOLUTIONS 

concentration  of  a  substance  is  meant  in  general  the  quantity  of  it  per 
unit-volume  either  of  the  solution  or  of  the  solvent.  Concentration 
expressed  in  equivalents  of  solute  per  liter  of  solution  (N/V)  is  called 
normal  concentration  (c).  This  form  of  concentration,  familiar  in 
volumetric  analysis,  will  be  used  in  this  book  in  connection  with  prop- 
erties, like  the  electrical  conductivity  of  solutions,  which  are  directly 
related  to  the  volume  of  the  solution.  Concentration  expressed  in  mols 
or  formula-weights  per  liter  is  called  molal  or  formal  concentra- 
tion* (c).  This  is  referred  sometimes  to  the  volume  (v)  of  the  solution, 
and  sometimes  to  that  of  the  solvent  (VQ~)  ;  but  throughout  this  book  it 
will  always  be  used  to  denote  the  number  of  mols  or  formula-weights  of 
solute  per  liter  of  solvent  (N/v0)  at  the  temperature  under  consid- 
eration. 

ProT).  19.  Calculate  at  25°  the  normal  concentration  and  the  formal 
concentration  (as  above  denned)  of  a  solution  containing  4.675%  H2SO4. 
The  density  at  25°  of  this  solution  is  1.0279,  and  that  of  water  is  0.9971. 

Henry's  law  may  now  be  stated  as  follows :  Any  chemical  substance 
present  in  a  gaseous  phase  and  in  a  solution  in  equilibrium  with  it  has 
at  any  definite  temperature  a  concentration  c  in  the  solution  which  is 
proportional  to  its  (partial)  pressure  p  in  the  gaseous  phase.  That  is, 

c/p  =  K, 

where  TL  is  an  equilibrium-constant  which  is  determined  by  the  nature 
of  the  chemical  substance  and  of  the  solvent  and  by  the  temperature. 
The  equilibrium  concentration  c  is  the  solubility  of  the  substance  when 
its  partial  pressure  in  the  gas  phase  is  p,  and  the  equilibrium-constant 
K  may  be  called  the  solubility-constant  of  the  gaseous  substance  in  the 
solvent.  Henry's  law  is  therefore  a  law  of  the  solubility  of  gases. 

This  second  form  of  Henry's  law  is  for  dilute  solutions  substan- 
tially equivalent  to  the  first  form.  For,  on  the  one  hand,  the  mol- 
fraclion  N/(N0-\-N),  as  it  approaches  zero,  becomes  equal  to  the 
mol-ratio  N/N0,  which  is  evidently  proportional  to  the  molal  concen- 
tration N/v0 ;  and,  on  the  other  hand,  the  vapor-pressure  of  a  substance 
in  a  solution  is  equal  to  its  partial  pressure  in  a  gaseous  phase  that  is 
in  equilibrium  with  the  solution. 

*The  number  of  mols  of  a  substance  present  in  a  solution  is  not  necessarily 
equal  to  the  number  of  formula-weights  of  it  added  ;  for  the  molecules  of  the 
substance,  owing  to  dissociation  or  association,  may  have  in  the  solution  a 
composition  different  from  that  represented  by  the  formula. 


VAPOR-PRESSURE  AND  BOILING-POINT  25 

Henry's  law  in  either  of  its  two  forms  is  conformed  to  more  closely 
as  the  pressure  of  the  gas  and  the  concentration  of  the  solute  ap- 
proach zero.  Like  the  other  laws  of  perfect  solutions,  it  usually  holds 
true  with  an  accuracy  of  2  to  3  percent,  even  when  the  pressure  is  one 
atmosphere  and  the  concentration  1  molal. 

It  is  to  be  noted  that  Henry's  law  expresses  conditions  of  equi- 
librium, and  that  these  conditions  are  often  attained  between  a  gaseous 
and  liquid  phase  only  by  long-continued  intimate  contact. 

From  a  molecular  standpoint,  Henry's  law,  like  Raoult's  law,  relates 
to  the  distribution  of  some  definite  kind  of  molecule  between  the  gas 
phase  and  the  liquid  phase.  Hence  in  applications  of  it  the  same 
chemical  substance  in  the  two  phases  must  be  considered.  Thus,  when 
the  chemical  substance  SO2  dissolves  in  water  it  is  largely  converted 
into  H2SO3  and  its  ions  H+  and  HSO3~;  and  Henry's  law  therefore 
requires,  not  that  the  total  concentration  of  solute  in  the  solution,  but 
that  the  concentration  of  the  S02  itself,  be  proportional  to  the  partial 
pressure  of  the  SO2  in  the  vapor.  When,  however,  the  only  change  in 
the  substance  is  that  it  partially  combines  with  the  solvent  forming  a 
solvate  (a  hydrate  in  the  case  of  water),  then  the  total  concentration 
may  be  employed;  for  the  fraction  solvated  is  in  dilute  solution  in- 
dependent of  the  concentration  of  the  substance,  as  may  be  shown 
by  the  mass-action  law.  Thus,  though  the  substance  CO2  on  dissolving 
in  water  is  partly  converted  into  the  hydrate  H2CO3  (which  is  sub- 
stantially unionized,  except  at  very  small  concentrations),  yet  the 
solubility  of  carbon  dioxide  gas,  as  found  by  determining  the  total 
quantity  dissolved,  changes  with  the  pressure  in  accordance  with 
Henry's  law. —  Correspondingly,  in  applying  Henry's  law  the  partial 
pressure  of  the  chemical  substance  in  the  gas  phase,  not  the  total 
pressure  of  the  gas,  must  be  considered.  Thus  the  quantity  of  carbon 
dioxide  dissolved  by  water  in  contact  with  air  is  not  determined  by  the 
pressure  of  the  air,  but  by  the  partial  pressure  of  CO2  in  the  air. 

Prob.  20.  A  mixture  of  air  and  ammonia  containing  1  mol-percent  of 
NH3  is  passed  at  25°  and  1  atm.  through  water.  The  saturated  solution  is 
found  by  titration  to  be  0.553  formal  in  NH4OH.  Calculate  the  partial 
vapor-pressure  of  NH3  in  a  1  formal  solution  at  25°.  The  vapor-pressure 
of  water  at  25°  is  23.8  mm. 

Prob.  21.  In  a  gas  burette  over  mercury  60  ccm.  of  dry  carbon  dioxide 


26  MOLAL  PROPERTIES  OF  SOLUTIONS 

at  25°  and  1  atm.  are  placed,  40  ccm.  of  water  are  introduced,  and  the  gas 
and  water  are  shaken  together  at  25°  till  equilibrium  is  reached,  keeping 
the  pressure  on  the  gas  1  atm.  The  volume  of  the  (moist)  gas  is  then 
found  to  be  28.9  ccm.  Calculate  the  molal  solubility  of  carbon  dioxide 
in  water  at  25°  when  its  partial  pressure  is  1  atm.,  neglecting  effects  that 
influence  the  result  less  than  0.5%. 

Pro&.  22.  At  20°  100  ccm.  water  dissolve  3.4  ccm.  of  oxygen,  1.7  ccm. 
of  nitrogen,  and  3.8  ccm.  of  argon  when  the  pressure  of  each  gas  is  1  atm. 
a.  Calculate  the  corresponding  solubility  of  each  gas  expressed  in  terms 
of  its  molal  concentration.  &.  Calculate  the  mol-fraction  of  each  con- 
stituent in  the  gas-mixture  obtained  by  shaking  water  with  air  at  20°, 
expelling  the  dissolved  gas  by  boiling  and  drying  it.  Tabulate  the  molal 
composition  of  this  gas  with  that  of  air  (given  in  Art.  15). 

Profc.  23. — Application  of  Henry's  Law  to  the  Determination  of  the 
State  of  Substances  in  Solution. — The  partial  vapor-pressure  of  NH3  in 
an  aqueous  solution  0.3  formal  in  ammonia  and  0.1  formal  in  AgNO3  is  at 
25°  equal  to  that  in  a  0.1  formal  solution  of  ammonia  in  water.  State 
and  explain  the  conclusion  that  can  be  drawn  as  to  the  formula  of  the 
complex  salt  formed,  considering  all  the  silver  nitrate  to  be  combined 
with  ammonia,  and  assuming  that  the  solubility-constant  of  the  NH3  is 
not  affected  by  the  silver  salt  in  the  solution. 

The  addition  to  the  water  of  a  salt  which  does  not  react  with  the 
volatile  solute  commonly  decreases  the  value  of  the  solubility-constant 
expressing  the  ratio  of  the  concentration  of  the  solute  in  the  solution 
to  its  pressure  in  the  gas-phase.  This  phenomenon,  which  is  known 
as  the  salting-out  effect,  is  subject  to  the  following  principles :  (1)  The 
decrease  of  the  solubility-constant  is  approximately  proportional  to  the 
concentration  of  the  added  salt,  up  to  concentrations  not  much  exceed- 
ing 1-normal.  (2)  The  fractional  decrease  per  equivalent  of  salt  per 
liter  is  roughly  the  same  for  a  definite  salt,  whatever  be  the  nature  of* 
the  solute.  (3)  This  fractional  decrease  varies  greatly  with  the  nature 
of  the  salt;  thus  the  decrease  caused  by  0.1  equivalent  of  salt  per  liter 
of  solution  varies  from  about  zero  in  the  case  of  barium  nitrate  to 
about  4%  in  the  case  of  potassium  and  sodium  sulphates. 

Profc.  24.  The  solubility  (that  is,  the  concentration  of  the  saturated 
solution)  of  carbon  dioxide  at  25°  and  1  atin.  is  0.0338  molal  in  pure 
water  and  0.0331  in  a  normal  NaCl  solution.  The  vapor-pressure  of 
ammonia  from  a  0.5  molal  solution  of  it  in  water  at  25°  is  6.65  mm. 
Predict  from  the  principles  of  the  salting-out  effect  the  ammonia  vapor- 
pressure  for  a  solution  0.5  molal  in  NH8  and  0.5  normal  in  NaCl. 


VAPOR-PRESSURE  AND  BOILING-POINT  27 

28.  Determination  of  Equilibrium-Conditions  by  the  Perpetual- 
Motion  Principle. 

Prob.  25.  A  volatile  substance  S  is  dissolved 
in  each  of  two  non-iniscible  solvents,  in  each  of 
which  it  has  the  same  molecular  weight  as  in  the 
gaseous  state.  The  two  solutions  A  and  B  are 
shaken  together  at  some  definite  temperature 
till  equilibrium  is  reached,  and  are  placed,  as  in 
Figure  2,  in  contact  with  the  vapor-phase  contain- 
ing the  substance  S  at  a  pressure  equal  to  its 

FIGURE  2 
partial  vapor-pressure  in  solution  A.  Prove  that  this 

pressure  must  also  be  the  partial  vapor-pressure  of  S  in  the  solution  B, 
by  showing  that,  if  it  were  greater  or  less,  the  substance  S  would 
pass  continuously  through  the  three  phases  of  the  system,  thus  pro- 
ducing perpetual  motion,  which  is  impossible. 

Perpetual  motion  (of  the  kind  involved  in  Prob.  25)  signifies  an 
ideal  process  by  which  an  unlimited  amount  of  work  might  be  pro- 
duced by  a  system  (i.  e.,  an  arrangement  of  matter)  operating  in 
surroundings  of  constant  temperature  and  drawing  from  them  no 
work.  Thus,  in  the  case  considered  in  Prob.  25,  if  the  vapor-pressures 
of  the  substance  in  the  two  solutions  were  different,  a  current  of  its 
vapor  would  flow  continuously  from  one  surface  to  the  other,  and 
work  could  be  obtained  from  the  moving  vapor  for  an  unlimited 
period  of  time,  for  example,  by  placing  a  windmill  in  the  vapor- 
space.  Even  though  a  quantity  of  heat  equivalent  to  the  work  pro- 
duced were  taken  up  from  the  surroundings,  the  process  would  still  be  a 
kind  of  perpetual  motion  which  is  impossible,  as  will  be  seen  later  "in 
the  discussion  of  the  second  law  of  energetics. 

The  principle  that  perpetual  motion  of  this  kind  is  impossible  is 
often  employed,  as  in  this  instance",  for  determining  the  conditions  of 
equilibrium  between  the  .different  phases  of  a  system.  It  leads  in 
such  cases  to  the  general  conclusion  that,  if  two  phases  are  each 
in  equilibrium  with  a  third  phase,  they  must  be  in  equilibrium  with 
each  other.  It  will  hereafter  be  called  simply  the  perpetual-motion 
principle. 

29.  Law  of  the  Distribution  of  a  Solute  between  Two  Non- 
Miscible  Solvents.  —  At   any  definite  temperature  the  ratio  between 
the  equilibrium  concentrations  of  a  chemical  substance  S  dissolved  in 


28  MOLAL  PROPERTIES  OF  SOLUTIONS 

two  non-miscible  solvents   (A  and  B)    is   constant,  whatever  be  the 
values  of  those  concentrations.    That  is, 


where  K  is  a  constant,  called  the  distribution-  ratio,  determined  by  the 
nature  of  the  substances  A,  B,  and  S,  and  by  the  temperature. 

This  law  may  be  generalized  so  as  to  be  applicable  to  the  dis- 
tribution of  a  definite  chemical  substance  between  any  two  kinds  of 
phases.  Thus  in  its  general  form  it  includes  Henry's  law,  since  the 
pressure  of  a  gas  at  any  definite  temperature  is  proportional  to  its  con- 
centration (that  is,  since  p  =  (N/v~)  R  T  =^  c  R  T).  It  is  called  the  law 
of  distribution  between  phases,  or  simply  the  distribution-law. 

This  distribution  law,  like  Raoult's  law  and  Henry's  law,  is  a  limit- 
ing law  which  becomes  more  exact  as  the  concentrations  approach  zero. 

The  distribution-ratio  of  a  solute  between  water  and  another  sol- 
vent is  decreased  by  the  addition  of  a  salt  to  the  water  in  accordance 
with  the  principles  of  the  salting-out  effect  stated  in  Art.  27. 

Prob.  26.  Derive  the  law  of  the  distribution  of  a  volatile  solute 
between  two  solvents  from  Henry's  law  and  the  conclusion  reached 
in  Prob.  25. 

Prob.  27.  At  25°  the  vapor-pressure  of  ammonia  above  a  0.1  inolal 
solution  of  it  in  chloroform  is  33.25  mm.,  and  above  a  0.5  molal  solu- 
tion of  it  in  water  is  6.65  mm.  a.  What  is  the  distribution-ratio  of 
ammonia  between  water  and  chloroform?  b.  What  would  be  its  dis- 
tribution-ratio between  a  0.5  normal  NaCl  solution  and  chloroform? 

Prob.  28.  The  distribution-ratio  of  an  organic  acid  between  water 
and  ether  at  20°  is  0.4.  A  solution  of  5  g.  of  the  acid  in  100  ccm.  water 
is  shaken  successively  with  three  20-ccrn.  portions  of  ether,  a.  How 
much  acid  is  left  in  the  water?  b.  How  much  acid  would  have  been 
left  in  the  water  if  the  solution  had  been  shaken  once  with  a  60-ccm. 
portion  of  ether? 

Prob.  29.  An  aqueous  solution  0.25  formal  in  KC1  and  0.20  formal 
in  HgCl2  is  shaken  with  an  equal  volume  of  benzene  at  25°.  The  ben- 
zene phase  is  found  by  analysis  to  contain  0.0057  mol  HgCl2  per  liter,  but 
no  KC1.  The  distribution-ratio  of  HgCl2  between  water  and  benzene  at  25° 
is  13.3.  a.  Calculate  the  total  concentration  of  mercuric  chloride  in  the 
aqueous  phase  and  the  concentration  of  the  part  of  it  which  is  combined 
with  the  potassium  chloride  (neglecting  the  salting-out  effect),  b.  The 
complex  salt  has  been  shown  by  other  measurements  to  be  KHgCl3.  Tab- 
ulate its  concentration  and  those  of  the  (uncombined)  KC1  and  HgCl2. 


VAPOR-PRESSURE  AND  BOILING-POINT  29 

VAPOR-PRESSURE  AND  BOILING-POINT  OF  CONCENTRATED  SOLUTIONS 

30.  Relation  between  the  Vapor-Pressure  and  Molal  Composition 
of  Perfect  Concentrated  Solutions. — Concentrated  solutions  may  be 
divided  for  purposes  of  consideration  into  two  groups  as  follows. 
One  group  consists  of  those  solutions  whose  formation  out  of  their  pure 
components  (when  these  are  liquid)  is  not  attended  by  any  considerable 
change  of  temperature  or  volume,  and  whose  properties  in  general  are 
approximately  the  sum  or  average  of  those  of  the  pure  components. 
The  characteristic  of  such  solutions  is  that  neither  component  exerts 
a  specific  influence  on  the  properties  of  the  other  component.  Such 
solutions  conform  approximately — more  closely  as  the  condition 
characterizing  them  is  more  nearly  fulfilled — to  the  laws  of  perfect 
solutions.  The  other  group  consists  of  those  solutions  whose  compo- 
nents exert  a  marked  influence  upon  one  another.  -For  these  solutions 
no  general  laws  are  known. 

In  the  case  of  solutions  belonging  to  the  first  of  these  groups  the 
vapor-pressure  of  each  component  conforms  approximately  to  Raoult's 
law.  In  other  words,  the  partial  vapor-pressure  of  each  component  is 
approximately  equal  io  the  product  of  its  mol-fraction  in  the  solution 
by  its  vapor-pressure  in  the  pure  statej  whatever  be  the  proportion  in 
which  the  components  are  present;  that  is,  PA  =  :?>OA#A,  PB  =  POB^B,  .  •  • 

The  method  commonly  employed  for  determining  the  partial  vapor- 
pressures  of  the  components  of  solutions  at  any  definite  temperature 
is  to  distil  off  a  small  fraction  from  a  large  volume  of  the  solution, 
adjusting  the  pressure  on  the  liquid  so  that  it  boils  at  this  temperature. 
The  composition  of  the  distillate  is  then  determined  by  chemical 
analysis  or  by  the  measurement  of  some  physical  property,  such  as 
density;  and  from  this  composition  and  the  pressure  under  which  the 
distillation  took  place  the  partial  vapor-pressures  are  calculated,  as 
illustrated  in  the  following  problem.  \j 

Prob.  30.  A  solution  of  two  substances  A  and  B  containing  ATA  mols 
of  A  and  A7B  mols  of  B  boils  at  the  temperature  T  when  a  pressure  of  p 
is  exerted  upon  it.  The  first  portion  of  distillate  consists  of  tfA'  mols  of 
A  and  <YB'  mols  of  B.  Derive  an  algebraic  expression  for  the  partial 
vapor-pressures  pA  and  pB  of  the  two  substances  in  the  solution,  ex- 
plaining the  principles  involved. 


CO  MOLAL  PROPERTIES  OF  SOLUTIONS 

Profc.  31.  At  50°  the  partial  vapor-pressures  of  benzene  and  of 
ethylene  chloride  in  solutions  of  these  two  substances  have  been  found 
experimentally  to  have  the  following  values  : 

Mol-fraction  Vapor-pressure  Vapor-pressure 

of  CaH6  of  CBHe  of  C.,H4Cl, 

1.000  268.0  mm.  0.0  mm. 

0.707  190.0  69.0 

0.478  128.0  124.0 

0.246  66.0  178.0 

0.000  0.0  236.0 

a.  Plot  on  a  large  scale  these  partial  vapor-pressures,  and  the  corre- 
sponding total  vapor-pressures,  as  ordinates  against  the  mol-fractions 
as  abscissas.  Show  that  the  three  graphs  are  in  almost  complete  accord 
with  Raoult's  law.  6.  Calculate  the  mol-fraction  of  benzene  in  the 
vapor  which  at  50°  is  in  equilibrium  with  each  of  the  three  solutions 
for  which  the  data  are  given  in  the  above  table,  c.  On  a  new  larger-scale 
vapor-pressure  composition  diagram  (with  ordinates  covering  only  the 
interval  of  230-270  mm.  )  draw  a  line  showing  the  variation  of  the  total 
vapor-pressure  with  the  mol-fraction  of  the  liquid  mixture.  Plot  also 
on  this  diagram  the  compositions  of  the  vapor  calculated  in  6  against 
the  total  vapor-pressures,  considering  that  the  abscissas  now  represent 
the  mol-fraction  of  benzene  in  the  vapor. 

Pro&.  32.  —  Distillation  at  Constant  Temperature.  —  At  50°  a  small 
fraction  is  distilled  off  from  a  large  volume  of  a  solution  containing 
equimolal  quantities  of  benzene  and  ethylene  chloride,  and  this  dis- 
tillate is  redistilled  at  50°.  Derive  from  the  diagram  of  Prob.  31  the 
mol-fraction  of  benzene  in  the  first  part  of  the  second  distillate. 

31.  Relation  between  the  Boiling-Point  and  Molal  Composition  of 
Perfect  Concentrated  Solutions.  —  When  a  solution  of  any  concentra- 
tion contains  only  one  volatile  component  and  the  vapor-pressure  of 
this  component  conforms  to  Raoult's  law,  an  exact  differential  expres- 
sion for  the  rise  in  boiling-point  dT  of  the  solution  produced  by  an 
increase  dx  in  the  mol-fraction  of  the  solute  can  be  obtained  by  combin- 
ing the  appropriate  equations  (as  in  Prob.  33).  This  expression  is: 


• 

l  —  x 

In  this  expression  Aif0  represents  the  molal  heat  of  vaporization  of  the 
pure  solvent  at  the  temperature  T.  The  expression  can  be  integrated 
(as  in  Prob.  34),  usually  without  significant  error,  under  the  assump- 
tion that  kH0  does  not  vary  within  the  temperature-interval  involved. 


VAPOR-PRE88URB  AND  BOILING-POINT  31 

The  boiling-point  can  also  be  determined  graphically  from  the 
vapor-pressure  curve  of  the  solvent  as  given  by  direct  measurements 
and  from  that  of  the  solution  as  computed  by  Raoult's  law,  as  was 
illustrated  in  Prob.  13. 

When  a  solution  contains  two  volatile  components  whose  partial 
vapor-pressures  both  conform  to  Raoult's  law,  its  boiling-point  can  best 
be  derived  graphically  from  the  vapor-pressures  of  the  pure  substances 
(as  in  Prob.  35). 

*Prob.  33. —  Boiling-Point  of  Perfect  Concentrated  Solutions  with 
One  Volatile  Component. — A  consideration  (like  that  in  Prob.  13)  of  the 
vapor-pressure  curves  shows  that  the  following  relation  holds  true  for 
any  solution  with  one  volatile  component,  whatever  be  its  concentration : 


dp/x\dx/T 

a.  State  in  words  what  each  of  these  partial  derivatives  signifies,  b.  Find 
from  the  approximate  Clapeyron  equation  (Art.  22)  and  from  the  Raoult 
equation,  respectively,  expressions  for  the  last  two  of  these  partial  de- 
rivatives, c.  Obtain  by  substituting  these  expressions  in  the  above  given 
equation  and  transforming  the  differential  equation  given  in  the  preced- 
ing text.  ( The  symbol  AIT ,  which  in  the  Clapeyron  equation  as  applied  in 
this  derivation  represents  the  quantity  of  heat  taken  up  from  the  sur- 
roundings when  1  mol  of  solvent  vaporizes  out  of  an  infinite  quantity  of 
the  solution  at  the  temperature  T,  can  be  shown  to  be  identical,  in  case 
the  solution  conforms  to  Raoult's  law,  with  the  rnolal  heat  of  vaporiza- 
tion of  the  pure  solvent  at  the  temperature  T.  It  can  also  be  shown,  by 
deriving  the  partial  derivatives  from  the  exact  Clapeyron  equation  and 
from  the  more  exact  Raoult  equation  (given  at  the  end  of  Art.  24),  that 
the  boiling-point  equation  given  in  the  preceding  text  holds  true  with 
substantial  accuracy  up  to  a  mol-fraction  of  solute  of  5%  to  10%,  even 
when  the  vapor  deviates  from  the  perfect-gas  law.) 

Prob.  34-  Integrate  the  equation  given  in  the  text,  assuming  that  the 
heat  of  vaporization  does  not  vary  with  the  temperature,  so  as  to  obtain 
a  relation  between  the  boiling-point  of  the  solution,  the  boiling-point  of 
the  pure  volatile  component,  and  its  mol-fraction  in  the  solution,  b.  Cal- 
culate the  boiling-point  of  a  solution  consisting  of  10  mols  of  a  non- 
volatile solute  and  90  mols  of  benzene.  The  heat  of  vaporization  of 
one  gram  of  benzene  at  its  boiling-point  80.3°  is  93.0  cal. 

Prob.  35.— Boiling-Point  of  Perfect  Solutions  with  Two  Volatile 
Components. — The  vapor  pressure  at..  80°,  83°,  86°,  89°,  92° 

of  a  pure  liquid  A  is 560,    610,    665,    725,   790  mm., 

and  of  a  pure  liquid  B  is 400,   435,   475,    520,    570  mm. 

*This  problem  may  be  omitted  in  briefer  courses  or  by  those  unfamiliar  with 
partial  derivatives. 


32  MOLAL  PROPERTIES  OF  SOLUTIONS 

a.  With  the  aid  of  these  data  and  Raoult's  law,  draw  on  a  large-scale 
vapor-pressure-composition  diagram  for  each  temperature  two  lines — one 
representing  the  total  vapor-pressure  of  any  solution  of  A  and  B;  and 
the  other  representing  the  partial  vapor-pressure  of  A  above  any  solu- 
tion. &.  Determine  from  the  plot  the  composition  of  the  liquid  which 
at  570  mm.  boils  at  each  of  these  temperatures ;  also  the  composition  of 
the  vapor  which  is  in  equilibrium  with  each  of  these  solutions  at  its 
boiling-point.  Determine  also  the  boiling-point  of  the  pure  liquid  A  at 
570  mm.,  and  tabulate  all  of  these  results,  c.  On  another  large-scale 
diagram  plot  against  these  liquid-compositions  the  boiling-points  as 
ordinates.  Plot  also  the  vapor-compositions  against  the  corresponding 
boiling-points. 

Distillation  at  Constant  Pressure  of  Perfect  Solutions  with  Two 
Volatile  Components. — 

Pro 6.  36.  A  solution  of  100  mols  of  each  of  the  liquids  A  and  B  of 
Prob.  35  is  distilled  at  570  mm.  until  its  boiling-point  rises  0.5°.  a.  Find 
from  the  diagram  of  Piob.  35c  the  molal  compositions  of  the  first  and  last 
portions  of  the  distillate.  &.  Regarding  the  composition  of  the  whole 
distillate  as  the  mean  of  that  of  its  first  and  last  portions  (which  is 
approximately  true  when  only  a  small  fraction  of  the  liquid  distils  over), 
calculate  the  number  of  mols  of  A  and  of  B  in  the  distillate  and  in  the 
residue,  c.  The  distillation  of  the  residue  is  continued  till  its  boiling- 
point  rises  0.5°  more.  Calculate  as  in  6  the  number  of  mols  of  A  and  B 
in  this  second  distillate  and  in  the  residue,  d.  Tabulate  the  number  of 
mols  of  A  and  of  B  in  the  original  liquid,  the  first  distillate,  the  second 
distillate,  and  the  final  residue;  the  mol-fraction  of  A  in  each  of  these 
liquids ;  and  the  boiling-point  of  each  of  them. 

Prob.  31.  a.  The  first  distillate  obtained  in  Prob.  36a  is  redistilled 
until  the  residue  attains  the  composition  of  the  second  distillate  obtained 
in  Prob.  36c.  Find  the  mol-fraction  of  A  in  the  new  distillate  and  its 
boiling-point.  6.  The  residue  is  now  mixed  with  the  second  distillate 
obtained  in  Prob.  36c,  and  the  distillation  is  continued  till  the  residue 
has  the  composition  of  the  residue  obtained  in  Prob.  36c.  Find  the  mol- 
fraction  of  A  in  the  distillate  thus  obtained  and  its  boiling-point 
c.  Tabulate  the  composition  and  the  boiling-point  of  the  original  equi- 
molal  solution  and  of  the  three  fractions  into  which  it  has  now  been 
resolved. 

Note. —  It  is  clear  from  the  diagram  of  Prob.  35c  that  any  perfect 
solution  submitted  to  distillation  resolves  itself  into  a  distillate  contain- 
ing a  larger  proportion,  and  into  a  residue  containing  a  smaller  propor- 
tion, of  the  more  volatile  component.  It  is  evident  that,  in  consequence 
of  this  behavior,  the  two  components  can  be  completely  separated  from 
each  other  by  repeated  fractional  distillation,  carried  out  as  illustrated 
by  Probs.  36  and  37. 


VAPOR-PRESSURE  AND  BOILING-POINT  33 

32.  Relation  between  the  Vapor-Pressure  and  Composition  of 
Concentrated  Solutions  in  General. — There  are  comparatively  few 
actual  solutions  which  fulfil  strictly  the  criterion  stated  in  Art.  30 
of  being  formed  out  of  their  components  without  any  change  of  tem- 
perature or  volume;  and  correspondingly,  comparatively  few  concen- 
trated solutions  conform  completely  to  Raoult's  law  of  perfect  solutions. 
The  law  is  therefore  to  be  regarded  as  a  limiting  law,  from  which  actual 
solutions  deviate  to  an  extent  which  is  as  a  rule  roughly  indicated 
by  the  magnitude  of  the  changes  of  temperature  and  volume  attend- 
ing the  mixing  of  the  components. 

The  data  given  below  illustrate  the  magnitude  of  the  deviations 
from  the  law  for  a  variety  of  solutions.  The  first  two  columns  of 
figures  show  the  change  of  temperature  and  the  percentage  change 
of  volume  which  result  when  equimolal  quantities  of  the  two  sub- 
stances at  the  same  temperature  are  mixed.  The  last  column  shows 
the  percentage  difference  between  the  observed  and  calculated  values 
of  the  total  vapor-pressure  of  the  equimolal  solution.  This  devia- 
tion of  the  total  pressure  corresponds  roughly  to  the  deviations  of 
the  partial  pressure  from  Raoult's  law  for  the  reason  that  the  partial 
pressures  of  the  two  components  deviate  from  the  law  in  the  same 
direction. 

No.  Components  mChanffe  °*  Ghan?e  °f  Deviation 

Temperature  Volume  of  Pressure 

1.  CH3OH  and  C2H6OH  _  0.10°  0.00%  +  0.1% 

2.  C6H6  and  C6HBCH8  —  0.45  -j-0.16  —  0.4 

3.  C2H6G2HSO2  and 

C2H6C3H5O2  —  0.02  -f  0.02  +  0.6 

4.  C6H6  and  C2H4C12  —  0.35  +0.34  -f  0.1 

5.  C6H6  and  C6H14  —  4.7  +0.52  -f  11. 

6.  (CH8)2CO  and   CHC18  +12.4  —0.23  —20. 

7.  C6H6  and  C2H5OH  —  4.2  0.00  +50. 

8.  H2O  and  C2H6OH  +  3.0  —2.56  +30. 

It  will  be  seen  from  the  table  that  there  is  a  parallelism  between 
the  changes  of  temperature  or  volume  and  the  deviations  from 
Raoult's  law.  Thus  with  the  first  four  pairs  of  substances,  where 
these  changes  are  small,  the  deviations  are  less  than  1%;  and  with 
the  last  four  pairs,  where  either  the  temperature  or  volume  change 
is  large,  the  deviations  are  so  great  that  Raoult's  law  cannot  be  said 
to  afford  even  a  rough  estimate  of  the  actual  values  of  the  pressure. 


34  MOLAL  PROPERTIES  OF  SOLUTIONS 

There  have  not  been  discovered  any  very  definite  relations  be- 
tween the  magnitude  of  the  deviations  and  the  nature  of  the  sub- 
stances; but  the  following  rules  furnish  useful  indications:  (1) 
Substances  which  are  closely  related  chemically,  such  as  the  neigh- 
boring members  of  the  same  homologous  series  (like  the  first  three 
pairs  of  substances  in  the  table),  or  those  which  differ  only  in  that 
they  contain  a  different  halogen  (like  chlorbenzene  and  brombenzene ) . 
conform  very  closely  to  Raoult's  law,  whether  or  not  the  molecules 
of  the  pure  substances  are  associated  in  the  way  described  in  the  last 
paragraph  of  Art.  26.  (2)  Substances  not  closely  related  chemically 
which  have  non-associated  molecules  sometimes  conform  very  closely 
to  Raoult's  law  (like  the  fourth  pair  in  the  table),  and  sometimes 
show  large  deviations  from  it  (like  the  fifth  and  sixth  pairs).  In 
most  cases  where  the  observed  pressure  is  less  than  the  calculated 
pressure,  these  deviations  are  probably  due  to  the  formation  of  a 
compound  between  the  two  substances  (as  has  been  shown  to  be  true 
of  the  sixth  pair  in  the  table).  In  those  cases  (like  that  of  the  fifth 
pair)  where  the  observed  pressure  is  greater  than  the  calculated 
pressure,  the  excess  of  pressure  is  probably  due  to  a  physical  effect 
arising  from  a  change  in  the  attractions  between  the  molecules. 
(3)  When  a  substance  with  associated  molecules  is  mixed  with  one 
with  unassociated  molecules,  the  vapor-pressure  of  the  solution  is 
much  greater  than  that  calculated  by  Raoult's  law,  as  is  illustrated 
by  the  data  for  ethyl  alcohol  and  benzene,  the  seventh  pair  in  the 
table.  This  effect  is  doubtless  due  in  part  to  the  breaking  down  of 
the  associated  molecules  of  the  one  substance  by  the  dilution  with 
the  other  substance,  whereby  the  partial  pressure  of  the  first  substance 
is  increased.  (4)  Substances  which  are  only  partially  miscible  (like 
ether  and  water)  form  solutions  whose  vapor-pressures  are  much  larger 
than  those  required  by  Raoult's  law. 

In  connection  with  these  deviations  it  should  be  borne  in  mind 
that,  as  the  mol-fraction  of  any  component  approaches  unity  (so 
that  the  solution  becomes  dilute  with  respect  to  the  other  component), 
its  partial  vapor-pressure  always  approaches  that  required  by  Raoult's 
law,  and  the  partial  vapor-pressure  of  the  other  component  conforms 
to  Henry's  law,  however  great  the  deviation  may  be  when  both  com- 
ponents are  present  in  large  proportion. 


VAPOR-PRESSURE  AND  BOILING-POINT  35 

Prob.  38.  Explain  by  reference  to  Raoult's  law :  a,  why  the  forma- 
tion of  a  non-volatile  compound  AB  between  the  two  components  A  and  B 
of  a  solution  causes  the  vapor-pressure  of  the  solution  to  be  less  than 
that  required  by  Raoult's  law;  ft,  why  a  substance  which  in  the  pure 
state  has  associated  molecules  may  have  an  abnormally  large  partial 
vapor-pressure  in  a  solution. 

Prob.  39.  a.  Draw  on  a  large-scale  diagram  vapor-pressure  curves 
representing  the  partial  vapor-pressure  at  35.2°  of  carbon  bisulphide  and 
of  acetone  in  solutions  of  these  components  throughout  the  whole  range 
of  composition.  In  accordance  with  the  principle  that  all  dilute  solu- 
tions conform  approximately  to  Raoult's  law  and  Henry's  law,  assume 
these  laws  to  hold  with  this  pair  of  components  up  to  5  mol-percent ; 
and  make  use  of  the  following  values  in  millimeters  of  the  vapor-pres- 
sures at  35.2° : 

Mol-percent  of  CS2  0         1      20      40      60      80       99    100 

Vapor-pressure  of  CS2  0     17.8    274     377    425    460  518 

Vapor-pressure  of  acetone  353  —  289  255  228  187  20.1  0 
I).  On  the  same  diagram  draw  a  curve  representing  the  total  vapor- 
pressures  of  the  solutions.  Draw  on  the  diagram  dotted  lines  showing 
what  the  partial  and  total  vapor-pressures  would  be  if  the  solutions 
behaved  as  perfect  solutions,  c.  Calculate  the  mol-percents  of  CS2  in  the 
vapor  in  equilibrium  with  the  5,  20,  40,  60,  80,  and  95  mol-percent  liquid 
solutions  ;  and  on  the  same  diagram  plot  these  vapor-compositions  against 
the  total  vapor-pressures  and  draw  a  dotted  line  through  the  points. 

Note. — With  the  aid  of  this  plot  the  behavior  of  any  solution  of  car- 
bon bisulphide  and  acetone  when  submitted  to  distillation  at  35.2°  could 
evidently  be  predicted. 

33.  Relation  between  the  Boiling-Point  and  Composition  of  Con- 
centrated Solutions  in  General. — The  boiling-point-composition  curves 
for  solutions  whose  vapor-pressures  do  not  conform  to  Raoult's  law 
can  be  based  only  upon  direct  experimental  determinations  of  the  boil- 
ing-points of  solutions  of  known  composition  and  upon  analyses  of 
the  corresponding  distillates,  All  that  can  be  done,  in  the  way  of 
generalization,  is  to  consider  the  different  types  of  curves  to  which 
different  pairs  of  substances  conform.  Figure  3  shows  the  three  types 
of  curves  exhibited  by  substances  miscible  in  all  proportions.  In  each 
case  the  solid  curve  shows  the  compositions  (expressed  as  mol-fractions) 
and  corresponding  boiling-points  of  the  liquid  solutions  at  one  atmos- 
phere ;  and  the  broken  curve  shows  the  composition  of  the  vapor  that  is 
in  equilibrium  with  these  solutions.  Thus  any  point  on  a  broken  curve 
represents  the  composition  of  the  vapor  of  the  liquid  solution  whose 
point  lies  in  the  same  horizontal  line. 


36 


MOLAL  PROPERTIES  OF  SOLUTIONS 


Curve  I  is  the  experimentally  determined  curve  for  solutions  of 
carbon  tetrachloride  (b.  pt.,  76.7°)  and  carbon  bisulphide  (b.  pt.,  46.3°). 
Curve  II  is  that  for  solutions  of  acetone  (b.  pt.,  56.2°)  and  chloroform 
(b.  pt.,  61.3°).  Curve  III  is  that  for  solutions  of  acetone  (b.  pt.,  56.2°) 
and  carbon  bisulphide  (b.  pt.,  46.3°). 

Solutions  of  type  I,  when  subjected  to  fractional  distillation,  behave 
like  perfect  solutions  (which  form  a  special  case  of  this  type),  and 
may  like  them  be  finally  resolved  into  the  pure  components.  The 
behavior  of  solutions  of  types  II  and  III  on  fractional  distillation  is 
shown  in  the  following  problems. 


\ 


\ 


\ 


\ 


\ 


1.0,      0.9 


0.8 


0.7 


0.5          0.4 
FIGURE  3 


0.2 


0.1 


VAPOR-PRESSURE  AND  BOIL1XG-POIXT  37 

Prob.  JfO.  a.  Determine  from  curve  III  in  Figure  3  the  boiling-point 
and  composition  of  the  first  portion  of  distillate  obtained  by  partial  dis- 
tillation of  solutions  of  acetone  and  carbon  bisulphide  containing 
70  mol-percent  of  acetone,  15  mol-percent  of  acetone,  and  35  mol-percent 
of  acetone,  respectively.  Tabulate  the  boiling-point  and  composition  of 
each  solution  and  distillate,  ft.  State  in  what  respects  each  distillate 
and  each  residue  differ  from  the  solution  from  which  it  was  obtained. 
o.  If  each  of  the  solutions  were  submitted  repeatedly  to  fractional  dis- 
tillation, what  would  be  the  composition  of  the  products  finally  obtained 
as  distillate  and  as  residue?  d.  If  of  the  70  mol-percent  solution  1000  g. 
were  so  fractionated,  what  weight  of  each  product  would  be  obtained? 

Pro 6.  41.  By  reference  to  curve  II  in  Figure  3  answer  the  same  ques- 
tions for  solutions  of  acetone  and  chloroform  as  are  asked  in  Prob.  55, 
a,  &,  and  c,  for  solutions  of  acetone  and  carbon  bisulphide. 

Just  as  the  solid  lines  in  such  temperature-composition  diagrams 
show  the  boiling-points  of  any  liquid  solution,  so  the  broken  lines  show 
the  condensation-point  at  one  atmosphere  of  vapor  of  any  composition ; 
the  composition  of  the  liquid  which  first  condenses  out  of  it  being  given 
by.  the  corresponding  point  on  the  solid  line.  Such  diagrams  therefore 
serve  to  predict  the  behavior  of  vapors  when  subjected  to  fractional 
condensation,  as  illustrated  by  the  following  problem. 

Proft.  4%-  d"  A  vapor  composed  of  equimolal  quantities  of  carbon 
tetrachloride  and  carbon  bisulphide  is  cooled  at  1  atm.  till  condensation 
begins.  By  referring  to  curve  I  in  Figure  3  find  the  temperature  at  which 
condensation  begins,,  and  the  composition  of  the  condensate.  6.  The 
vapor  is  gradually  cooled,  removing  the  condensate  as  it  forms,  till  the 
temperature  falls  to  60°.  Find  the  composition  of  the  condensate  which 
is  now  separating,  and  that  of  the  residual  vapor,  c.  Tabulate  the  com- 
position (50  mol-percent)  of  the  original  vapor,  the  average  composition 
of  the 'condensate  obtained  from  it  in  6,  and  the  composition  of  the  resid- 
ual vapor.  Include  in  the  table  also  the  composition  of  the  liquid  which 
upon  distillation. would  furnish  the  original  vapor. 

The  processes  used  in  chemical  practice  for  the  separation  of  vola- 
tile liquids,  such  as  alcohol  and  water,  involve  fractional  distillations 
and  condensations  taking  place  in  accordance  with  the  principles  here 
considered. 


38  HOLAL  PROPERTIES  OF  SOLUTIONS 

FREEZING-POINT    OF    SOLUTIONS 

34.  Freezing-Point   and   Its   Relation  to   Vapor-Pressure. — The 
freezing-point  of  a  liquid  is  that  temperature  at  which  the  solid  solvent 
and  the  liquid  coexist  in  equilibrium  with  each  other.    The  solid  which 
separates  from  the  solution  commonly  consists  of  the  pure  solvent; 
and  this  is  assumed  to  be  the  case  throughout  the  following  consider- 
ations.   In  any  such  case  the  freezing-point  of  a  solvent  is  lowered  by 
dissolving  another  substance  in  it. 

Profc.  43.  Prove  that  at  the  freezing-point  of  a  solution  its  vapor- 
pressure  and  that  of  the  solid  which  separates  from  it  must  be  equal, 
by  showing  that  otherwise  perpetual  motion  would  result. 

Prob.  44-  The  vapor-pressures  of  ice  and  (supercooled)  water  be- 
tween 0°  and  — 4°  are  as  follows  : 

0°         —1°       —2°       —3°         — 4° 

Water 4.58        4.26        3.96        3.68        3.41  mm. 

Ice 4.58        4.22        3.89        3.59        3.30mm. 

a.  Calculate  by  Raoult's  law  the  vapor-pressure  at  each  of  these  tem- 
peratures of  a  solution  consisting  of  3  mols  of  solute  and  97  mols  of 
solvent.  Plot  on  a  diagram  the  vapor-pressures  of  this  solution,  of  water, 
and  of  ice  as  ordinates  against  the  temperatures  as  abscissas,  using  a 
scale  large  enough  to  enable  0.001  mm.  to  be  estimated  (by  including  on 
it  pressures  ranging  only  from  3.30  to  4.60  mm.).  6.  Determine  from  the 
plot  the  freezing-point  of  the  solution. 

With  the  aid  of  a  diagram  like  that  of  the  preceding  problem,  but 
including  also  the  vapor-pressure  curves  of  a  second  solution,  it  can  be 
shown  from  the  geometrical  relations  that  for  dilute  solutions  the 
lowering  of  the  freezing-point  is  proportional  to  the  lowering  of  the 
vapor-pressure  of  the  solvent  at  its  freezing-point.  It  can  also  be  shown 
that  the  proportionality-constant  is  dependent  on  the  difference  in  the 
slopes  of  the  vapor-pressure  curves  for  the  solid  solvent  and  for  the 
liquid  solvent. 

35.  Relation  between  Freezing-Point-Lowering  and  Molal  Com- 
position.— From  Raoult's  law  of  vapor-pressure  lowering  and  from 
the  Clapeyron  equations  expressing  the  change  of  the  vapor-pressure 
of  the  solid  and  the  liquid  solvent  with  the  temperature,  the  follow- 
ing equation  can  be  derived  by  a  method  similar  to  that  used  in  Arts.  25 


FREEZING-POINT  39 


and  26  for  obtaining  the  corresponding  expression  for  the  boiling-point 
raising: 

XT  T>  r\     _ 


N         RT<?    N 


rp       _    rn  -    p  _  •__    - 

No  ~  '  A#0  No 
In  this  equation  T0  is  the  freezing-point  of  the  solvent,  T  that  of 
a  solution  of  TV  mols  solute  in  N0  mols  solvent,  B  a  quantity  char- 
acteristic of  the  solvent  called  its  freezing-point  constant,  and  A//0  the 
heat  absorbed  by  the  fusion  of  one  mol  of  the  solvent  at  T0. 

In  place  of  this  freezing-point  constant,  the  molal  freezing-point 
lowering,  defined  analogously  to  the  molal  boiling-point  raising,  is 
commonly  recorded  in  chemical  literature. 

Prod.  Jf5.  When  one  gram  of  ice  at  0°  melts,  the  heat  absorbed 
is  79.7  cal.  a.  What  is  the  freezing-point  constant  for  water?  fc.  What 
is  its  molal  freezing-point  lowering? 

Prob.  46.  A  solution  of  0.60  g.  acetic  acid  in  50.0  g.  water  freezes 
at  —0.376°.  A  solution  of  2.32  g.  acetic  acid  in  100  g.  benzene  freezes 
0.970°  lower  than  pure  benzene.  The  freezing-point  constant  for  benzene 
is  65.4.  Calculate  the  molecular  weight  of  acetic  acid  in  each  of  these 
solvents,  and  state  what  the  results  show  in  regard  to  its  molecular 
formula  in  each  solvent. 

The  following  differential  expression  for  the  freezing-point  of  per- 
fect solutions  of  any  concentration  may  be  obtained  by  formulating  the 
appropriate  equations  : 

RT*      dx 


(1  -  x) 

In  this  equation  —  dT  denotes  the  lowering  produced  in  the  freezing- 
point  T7  of  a  solution  by  increasing  the  mol-fraction  x  of  the  solute 
by  dx,  and  &H0  denotes  the  heat  absorbed  by  the  fusion  of  one  mol  of 
the  solid  solvent  at  T.  This  equation  is  exact  at  any  concentration 
up  to  which  Kaoult's  law  holds  true.  It  may  be  integrated,  commonly 
without  significant  error,  under  the  assumption  that  A-ET0  does  not  vary 
within  the  temperature-interval  involved. 

The  freezing-point  of  other  concentrated  solutions,  for  which  there 
is  no  general  law,  cannot  be  calculated  from  the  molal  composition. 
The  relation  between  freezing-point  and  molal  composition  can,  of 
course,  be  experimentally  determined;  and  the  results  may  be  clearly 
represented  by  freezing-point-composition  diagrams,  analogous  to  the 
boiling-point-composition  diagrams.  Such  diagrams  are  considered 
in  Chapter  VII. 


40  MOLAL  PROPERTIES  OF  SOLUTIONS 

OSMOTIC    PRESSURE    OP    SOLUTIONS 

36.  Osmotic  Pressure. — When  a  solution  S  is  separated  from  a 
pure  solvent  W,  as  illustrated  in  Figure  4,  by  a  wall  aa  which  allows 
this  solvent  to  pass  through  it,  but  prevents  entirely  the  passage  of 
the  solute,  the  solvent  is  drawn  through  the 
wall  into  the  solution.  This  flow  of  solvent 
may,  however,  be  prevented  by  exerting  a 
pressure  p2  on  the  solution  greater  by  a 
definite  amount  than  the  pressure  pv  upon 
the  pure  solvent;  and  the  solvent  may  be 


w 


S 


forced  out  of  the  solution  by  exerting  a  still 

greater   pressure   upon   the    solution.      The  __ 

difference  of  pressures  on  solution  and  sol- 

FIGURE  4 

vent  which  produces  a  condition  of  equilib- 
rium such  that  there  is  no  tendency  of  the  solvent  to  flow  in  either 
direction  is  called  the  osmotic  pressure  P  of  the  solution. 

Walls  of  the  kind  just  described  are  known  as  semipermeable 
walls.  Certain  animal  membranes,  such  as  parchment  or  bladder, 
are  permeable  for  water,  but  not  for  certain  solutes  of  high  molecular 
weight  (the  so-called  colloids).  The  walls  of  some  animal  and  plant 
cells  are  very  perfect  semipermeable  walls.  The  most  satisfactory 
artificial  semipermeable  walls  have  been  made  by  precipitating  copper 
ferrocyanide  within  the  pores  of  an  unglazed  porcelain  cell,  which 
gives  the  precipitate  sufficient  rigidity  to  withstand  high  osmotic 
pressures.  The  cell,  filled  with  the  solution  and  immersed  in  pure 
water,  is  connected  with  a  manometer  whose  mercury  column  is  in 
direct  contact  with  the  solution.  By  means  of  such  cells  exact 
measurements  of  the  osmotic  pressure  of  aqueous  solutions  of  cane- 
sugar  and  glucose  have  been  made  up  to  pressures  of  25  atmospheresL 

Osmotic  pressure  plays  a  very  important  part  in  the  physiological 
processes  taking  place  in  the  bodies  of  animals  and  plants.  In  the 
study  of  the  general  principles  of  chemistry,  it  is  of  value  because  it 
is  a  property  which,  like  vapor-pressure,  enables  the  various  properties 
of  solutions  to  be  correlated  and  their  energy  relations  to  be  treated 
upon  the  basis  of  one  simple  fundamental  concept.  The  correspond- 
ence shown  in  Art.  38  to  exist  between  the  laws  of  the  osmotic  pressure 
of  dilute  solutions  and  those  of  the  pressure  of  perfect  gases  makes 
possible,  moreover,  a  closely  analogous  treatment  of  these  two  states. 


OSMOTIC  PRESSURE  41 

37.  Relation  of  Osmotic  Pressure  to  Vapor-Pressure. —  In  the 
osmotic  arrangement  represented  in  Figure  4,  when  the  pressures  Ft 
and  F2  are  such  that  there  is  equilibrium  and  therefore  no  tendency 
for  the  solvent  to  pass  through  the  semipermeable  wall,  it  follows  from 
the  perpetual  motion  principle  that  the  vapor-pressure  of  the  solvent 
in  the  pure  state  must  be  equal  to  its  vapor-pressure  in  the  solution. 
For,  if  a  vapor  phase  were  in  contact  (through  walls  permeable  only 
for  the  vapor)  both  with  the  solvent  and  the  solution,  there  would  evi- 
dently be  a  continuous  flow  of  the  solvent-substance  through  the  vapor- 
phase  and  back  through  the  wall  between  the  two  liquid  phases,  unless 
its  vapor-pressures  in  these  liquid  phases  were  equal.  That  this  condi- 
tion is  realized  results  from  the  fact  that  the  vapor-pressure  of  a  liquid 
is  increased  by  increasing  the  pressure  upon  it.  Thus  the  larger  pres- 
sure P2  on  the  solution  increases  its  vapor-pressure  up  to  the  value  of 
the  vapor-pressure  of  the  solvent  under  the  smaller  pressure  Pr  From 
a  quantitative  consideration  of  the  effect  of  pressure  on  vapor-pressure 
a  relation  between  the  vapor-pressure  ,and  osmotic  pressure  of  solutions 
can  be  derived,  as  shown  in  the  following  problems. 

Effect  of  Pressure  on  the  Vapor-Pressure  of  Liquids. — 
*Prol).  47-  Consider  a  column  of  a  pure  solvent  contained  in  a  porous 
tube  impermeable  to  the  liquid  solvent,  but  permeable  to  its  vapor ;  and 
consider  this  tube  to  be  surrounded  by  the  vapor  of  the  solvent;  the 
whole  system  being  at  a  constant  temperature  T.  When  there  is  equi- 
librium the  vapor-pressure  of  the  solvent  at  any  level  must  evidently  be 
equal  to  the  pressure  of  the  vapor  at  that  level.  The  pressure  of  the 
liquid  or  of  the  vapor  must,  however,  be  greater  at  a  lower  than  a  higher 
level  by  the  (hydrostatic)  pressure  of  the  intervening  column  of  liquid 
or  vapor,  a.  Formulate  an  expression  for  the  increase  dP  of  the  pres- 
sure P  of  the  liquid,  and  one  for  the  increase  dp  of  the  pressure  p  of 
the  vapor,  corresponding  to  a  decrease  ( — dh)  in  the  height  h.  6.  By 
combining  these  expressions  and  making  appropriate  substitutions, 
derive  the  fundamental  equation  dp/dP  —  v0/v,  in  which ji?n  is  the  volume 
of  any  definite  weight  of  the  liquid  at  the  pressure  P  and  temperature  JC, 
and  t^  is  the  volume  of  the  same  weight  of  the  vapor  at  the  pressure  p 
and  temperature  T.  c.  Integrate  this  equation  between  the  limits  plt  Pu 
and  p,,  P2,  assuming  that  the  vapor  conforms  to  the  perfect-gas  laws  and 
that  the  volume  of  the  liquid  does  not  vary  with  the  pressure  upon  it. 

*Pro6.  48.    Find  the  ratio  of  the  vapor-pressure  of  water  at  4°  and 
10  atm.  to  that  at  4°  and  the  pressure  of  the  saturated  vapor. 

*  Problems  with  asterisks  attached  are  not  essential  to  the  subsequent  con- 
siderations and  may  be  omitted  in  brief  courses  on  the  subject. 


4lA  MOLAL  PROPERTIES  OF  SOLUTIONS 

Relation  of  Osmotic-Pressure  to  Vapor-Pressure. — 

*Pro&.  49.  a.  How  much  must  the  pressure  on  water  be  reduced  in 
order  that  it  may  be  in  equilibrium  through  a  sernipermeable  wall  with 
an  aqueous  solution  which  has  at  25°  a  vapor-pressure  98%  as  great  as 
that  of  water?  (The  water  must  evidently  be  subjected  to  a  negative 
pressure  or  suction.  With  proper  precautions  liquids  have  been  subjected 
to  negative  pressures  of  several  atmospheres,  without  the  column  break- 
ing.) 6.  What  is  the  osmotic  pressure  of  this  solution? 

The  preceding  problems  show  that  the  osmotic  pressure  P  at  the 
temperature  T  of  any  solution  consisting  of  N  mols  of  solute  and  N0 
mols  of  solvent  (whose  volume  in  the  pure  state  is  v0)  is  given  by  the 
expression : 

N0RT         po. 
P  =  -         -   log  - 

VQ  p 

In  this  expression  p0  represents  the  vapor-pressure  of  pure  solvent  and 
p  that  of  the  solvent  in  the  solution.  The  expression  is  substantially 
exact  even  at  large  concentrations,  provided  the  vapor  conforms  to  the 
perfect-gas  laws. 

Proft.  50.  At  100°  the  vapor-pressure  of  a  solution  consisting  of 
28  g.  NaCl  and  100  g.  H2O  is  624  mm.  What  is  its  osmotic  pressure? 
The  specific  volume  of  water  at  100°  is  1.043. 

38.    Relation  between  Osmotic  Pressure  and  Molal  Composition. — 

For  a  solution  whose  vapor-pressure  conforms  to  Raoult's  law,  the 
equation  derived  in  Art.  37  can  be  transformed  into  the  following  one : 

i        '       •     N' 
log 


**• 

For  dilute  solutions,  for  which  N  /N0  is  small,  this  equation  can  be 
simplified  by  expanding  the  logarithmic  term  into  a  series,  as  shown 
in  Prob.  51,  to  the  following  expression: 

or  P  =  cRT. 


In  these  expressions  P  is  the  osmotic  pressure  at  the  temperature  T 
of  a  solution  consisting  of  N  mols  of  solute  and  ^0  mols  of  solvent 
whose  volume  in  the  pure  state  is  t>0,  c  is  the  molal  concentration,  and 
R  is  the  gas-constant. 

The  osmotic  pressure  of  concentrated  solutions  to  which  Raoult's 
law  does  not  apply  cannot  be  calculated  from  the  composition  of  the 
solution.  It  may,  however,  be  determined  not  only  by  direct  measure- 
ment, but  also  from  vapor-pressure  measurements,  as  shown  in  Art.  37. 


REVIEW  PROBLEMS  4lB 

Prod.  51.  Derive  the  two  equations  given  in  the  preceding  text. 

Prob.  52.  The  lower  end  of  a  vertical  tube  is  closed  with  a  semi- 
permeable  wall  and  is  dipped  just  beneath  the  surface  of  a  pure  solvent. 
A  0.1  molal  solution  of  cane-sugar  (C^H^O^)  of  density  1.014  is  poured 
into  the  tube  until  the  hydrostatic  pressure  at  the  semipermeable  wall 
is  sufficient  to  prevent  water  from  entering  the  solution.  The  temper- 
ature is  4°.  What  is  the  height  of  the  column  in  meters? 


REVIEW  OF  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

39.     Review  Problems. 

Prod.  53.  a.  Summarize  the  equations  expressing  for  dilute  solutions 
the  approximate  relations  between  molal  composition  and  (1)  vapor- 
pressure,  (2)  boiling-point,  (3)  freezing-point,  (4)  osmotic  pressure. 
State  explicitly  what  each  symbol  signifies. 

Pro'b.  54-  Summarize  the  corresponding  equations  which  hold  true 
for  perfect  solutions  of  any  concentration. 

Prod.  55.  a.  Calculate  by  the  laws  of  dilute  solutions,  using  the 
data  given  below,  the  fractional  vapor-pressure-lowering,  the  boiling- 
point-raising,  the  freezing-point-lowering,  and  the  osmotic  pressure  at 
5.5°  of  a  solution  A  containing  0.1  mol  of  a  nonvolatile  solute  S  in  1000  g. 
benzene.  d.  Calculate  by  the  laws  of  perfect  concentrated  solutions 
the  values  of  the  same  quantities  for  a  solution  B  containing  2  mols 
of  the  solute  S  in  1000  g.  benzene.  Calculate  the  ratio  of  each  of  these 
values  to  the  corresponding  one  for  solution  A.  (Note  that  the  value  of 
this  ratio  would  be  20.0,  if  the  equations  for  dilute  solutions  were 
applicable  to  solution  B.)  c.  If  the  solute  in  solution  B  were  volatile 
and  at  the  boiling-point  of  the  solution  had  a  partial  pressure  of  20  mm., 
what  would  be  the  boiling-point  of  the  solution?  Data:  The  heat  of 
fusion  of  one  gram  of  benzene  at  its  freezing-point,  5.5°,  is  30.2  cal.  The 
heat  of  vaporization  of  one  gram  of  benzene  at  its  boiling-point,  80°,  is 
93.0  cal.  The  density  at  the  freezing-point  is  0.895. 

Prod.  56.  a.  Carbon  bisulphide  boils  at  46°  at  1  atm.  Its  molal 
heat  of  vaporization  at  this  temperature  is  6430  cal.  What  is  its 
boiling-point  constant?  d.  A  solution  of  15.5  g.  phosphorus  (at.  wt.. 
31.0)  in  1000  g.  CS2  boils  0.300°  higher  than  pure  carbon  bisulphide. 
What  is  the  molecular  weight  and  what  is  the  molecular  formula  of 
phosphorus  in  this  solvent? 

Prod.  57.  If  127  g.  iodine  (at.  wt.,  127)  were  added  to  the  solution 
of  Prob.  566,  how  much  higher  than  the  boiling-point  of  pure  carbon 
bisulphide  would  that  of  the  solution  be :  a,  in  case  the  iodine  remained 
uncombined  in  the  form  of  I2 ;  6,  in  case  it  all  combined  with  the  phos- 
phorus forming  P4I8;  c,  forming  P2I4;  d,  forming  PI2? 


41c  MOLAL  PROPERTIES  OF  SOLUTIONS 

Prol).  58.  Human  blood  freezes  at  — 0.56°.  What  is  its  osmotic 
pressure  at  37°? 

Prol).  59.  At  25°  the  distribution-ratio  of  Br2  between  carbon  tetra- 
chloride  and  water  is  38 ;  and  the  pressure  of  bromine  above  a  0.05  molal 
solution  of  Br2  in  water  is  50  mm.  If  one  liter  of  this  solution  be 
shaken  with  50  ccm.  carbon  tetrachloride,  what  will  be  the  pressure 
of  the  bromine  over  the  carbon-tetra chloride  phase?  (Bromine  exists 
in  all  three  phases  only  as  Br2.) 

Pro  6.  60.  When  ethyl  acetate  and  water  are  shaken  together  at 
20°,  two  liquid  phases  result,  one  phase  containing  1.75  mol-percent. 
and  the  other  86.8  mol-percent  ethyl  acetate,  a.  Show  that,  even  though 
Raoult's  law  may  hold  for  one  component  in  one  phase,  it  cannot  hold 
for  the  same  component  in  the  other  phase.  o.  Calculate  the  partial 
vapor-pressures  of  ethyl  acetate  and  water  over  the  mixture,  assuming, 
since  the  solutions  are  moderately  dilute,  that  Raoult's  law  applies  to  the 
solvent  in  each  phase.  At  20°  the  vapor-pressure  of  pure  ethyl  acetate 
is  73  mm.,  and  that  of  water  is  17.4  mm.  c.  Calculate  the  partial 
vapor-pressures  of  the  two  components  in  a  solution  containing  1.00 
mol-percent  of  ethyl  acetate. 

Prol).  61.  Aniline  (C6H3NH2)  is  distilled  with  steam  at  1  atm.  The 
vapor-pressure  of  aniline  is  46  mm.  at  100°  and  40  mm.  at  97°.  Water  and 
aniline  have  limited  solubilities  in  each  other;  one  phase  containing  at 
100°  1.5  mol-percent  of  aniline,  and  the  other  68  mol-percent  of  aniline. 
Find  the  boiling-point  of  the  mixture,  and  the  number  of  grams  of 
aniline  distilling  over  with  each  gram  of  water.  Assume  that  the  solu- 
bilities do  not  change  appreciably  within  the  small  temperature  interval, 
and  that  the  vapor-pressure  of  the  aniline  in  the  68  mol-percent  solution 
is  lowered  only  half  as  much  as  Raoult's  law  requires.  (This  last  assump- 
tion is  a  rough  estimate  based  on  the  considerations  that  the  partial 
vapor-pressure  of  the  aniline  in  the  68  mol-percent  solution  is  less  than 
that  of  pure  aniline  and  that,  in .  accordance  with  statement  (4)  on 
page  34,  it  is  greater  than  that  required  by  Raoult's  law.  In  the  absence 
of  more  definite  knowledge  it  is  obviously  best  to  assume  that  the  actual 
vapor-pressure  lies  midway  between  these  two  limiting  values.) 

Prol).  62.  At  1  atm.  pure  nitric  acid  has  a  boiling-point  of  86°.  A 
solution  of  nitric  acid  and  water  of  the  composition  HNO8  -f  1.6  H2O 
distils  at  1  atm.  at  a  constant  temperature  of  121°.  Make  a  diagram 
showing  the  character  of  the  boiling-point-composition  curve  for  this 
pair  of  substances.  Draw  in  on  the  diagram  a  curve  representing  in  a 
general  way  the  composition  of  the  vapor  in  equilibrium  with  any  solu- 
tion at  its  boiling-point.  State  what  products  would  finally  result  as 
distillate  and  residue  from  the  fractionation  of  the  three  solutions, 
HN08  +  H20,  HNO,  -f  1.6  H2O,  HNO8  +  3  H2O. 

Prob.  63.  Upon  partial  distillation  at  constant  pressure  a  solution 
of  two  components  A  and  B  containing  20%  A,  having  a  boiling-point  of 


REVIEW  PROBLEMS  41o 

80°,  yields  a  residue  containing  15%  A;  and  a  mixture  containing 
75%  A,  having  a  boiling-point  of  60°,  yields  a  distillate  containing  70%  A. 
Draw  a  diagram  which  will  show  in  a  general  way  the  character  of  the 
liquid-composition  and  of  the  vapor-composition  curve.  Predict  what 
products  would  finally  result  as  distillate  and  residue  from  the  fraction- 
ation  of  the  20%  and  of  the  75%  solution. 


CHAPTER   IV 

PROPERTIES   OF   SOLUTIONS  RELATED   TO   IONIC 
COMPOSITION 


EFFECT  OF  IONIZATION  ON  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

40.  Abnormal  Effects  of  Salts  on  the  Vapor-Pressure  and  Related 
Properties  of  Water,  and  their  Explanation  by  the  Ionic  Theory.— 

In  dilute  solution  the  effect  of  salts  of  the  uniunivalent  type,  such  as 
sodium  chloride  or  silver  nitrate,  on  the  vapor-pressure  of  water  and 
on  the  other  related  properties  is  nearly  twice  as  great,  and  the  effect 
of  salts  of  the  unibivalent  type,  such  as  potassium  sulphate  and  barium 
chloride,  is  nearly  three  times  as  great,  as  it  would  be  if  each  formula- 
weight  yielded  a  single  mol  in  the  solution.  Strong  acids  and  bases 
show  a  similar  behavior.  These  and  other  facts  have  led  to  the  con- 
clusion that  these  substances  are  largely  dissociated  in  aqueous  solu- 
tion. For  example,  NaCl  dissociates  into  Na+  and  Cl~;  HNO3  into 
IT  .and  NO3-;  K2SO4  into  K+,  K+,  and  SO4";  and  Ba(OH)2  into  Ba+\ 
OH~,  and  OH~.  Tri-ionic  substances  also  dissociate  partially  into 
intermediate  ions;  thus  I^SO,  into  KS04~  (and  K+),  and  Ba(OH)a 
into  BaOH+  (and  OH"). 

The  electrical  behavior  of  the  solutions  indicates  that  these 
dissociation-products  differ  from  ordinary  substances  in  that  their 
molecules  are  electrically  charged.  These  charged  substances  are 
called  ions;  and  to  their  formulas  +  or  —  signs  are  attached,  as  in 
the  above  examples,  to  indicate  the  nature  and  magnitude  of  the 
charge.  The  fraction  of  the  salt  dissociated  is  called  its  ionization  7. 
This  fraction  always  decreases  with  increasing  concentration. 

Prol).  1.  A  solution  of  0.65  formula- weight  KC1  in  1000  g.  water  has 
at  100°  a  vapor-pressure  of  744.8  mm.  Calculate  the  number  of  mols  of 
solute  per  formula-weight  of  salt  and  the  ionization  of  the  salt. 

Pro 6.  2.  According  to  an  estimate  based  upon  certain  of  its  proper- 
ties, sulphuric  acid  in  a  solution  containing  0.05  formula-weights  H2SO4 
per  1000  g.  water  at  0°  is  58%  ionized  into  H*  and  HSO4~  and  36% 
into  H+,  H+,  and  SO4".  Calculate  the  total  number  of  mols  of  solute  per 
formula-weight  of  the  acid  and  the  freezing-point  of  the  solution.  (The 
observed  freezing-point  is  — 0.215°.) 

42 


FARADAY'S    LAW  43 

CONDUCTION    OP    ELECTRICITY     IN    SOLUTIONS:    FARADAY'S    LAW 

41.  Electrolytic   Conduction.  —  Conductors  are  divided  into  two 
classes  with  reference  to  the  changes  that  are  produced  in  them  by 
the  passage  of  electric  currents.     Those  which  undergo  no  changes 
except  such  as  are  produced  by   a   rise   in   temperature   are  called 
metallic   conductors.     Those   in  which  the  passage  of  a   current   is 
attended  by  a  chemical  change  are  called  electrolytes.    Aqueous  solu- 
tions of  salts,  bases,  and  acids,  and  melted  salts  at  high  temperatures, 
are  the  most  important  classes  of  the  well-conducting  electrolytes. 
The  most  obvious  chemical  changes  attending  the  passage  of  a  current 
through  an  electrolyte  are  those  that  take  place  at  the  surfaces  of  the 
metallic  conductors  where  the  current  enters   and  leaves  the  elec- 
trolyte.   The  production  of  such  chemical  changes  by  a  current  from 
an  external  source   is  called   electrolysis.     The  occurrence   of  such 
changes,  when  they  themselves  give  rise  to  an  electric  current,  is  called 
voltaic  action.    Those  portions  of  the  metallic  conductors  that  are  in 
contact  with  the  electrolyte  are  called  the  electrodes;  the  one  at  which 
the  current  leaves  the  electrolyte,  or  the  one  towards  which  the  positive 
electricity  flows  through  the  electrolyte,  is  designated  the  cathode; 
the  other,  the  anode. 

42.  Chemical  Changes  at  the  Electrodes.—  The  chemical  change 
produced  at  the  cathode  is  always  a  reduction;  that  at  the  anode,  an 
oxidation.    The  products  resulting  when  aqueous  solutions  of  certain 
typical  salts,  bases,  and  acids  are  electrolyzed  between  electrodes  which 
are  unattacked  are  as  follows : 

Solute  Cathode  products  Anode  products 

Cu(N08)2  or  AgNO,  Cu  or  Ag  O2  and  HNO8 

KNO,  H2  and  KOH  O2  and  HNO3 

Na2SO4  H2  and  NaOH  O2  and  H2SO4 

KOH  or  Ba(OH)2  H,  Oa 

H2SO4  or  H8PO4  Ha  O2 

Dilute  HC1  or  HNO,  H,  O2 

Concentrated  HC1  Ha  Cla 

Concentrated  HNO,  NO2  and  NH,  Oa 

When  the  anode  is  a  metal  which  can  react  with  the  anion  of  the 
solute,  the  change  at  the  anode  consists  only  in  the  dissolving  of  the 
metal;  thus  when  a  nitrate  or  sulphate  is  electrolyzed  with  a  copper 
anode,  copper  passes  into  solution  forming  copper  nitrate  or  sulphate. 
In  the  case  of  voltaic  actions,  the  chemical  changes  are  of  a 


44  IONIC    PROPERTIES    OF    SOLUTIONS 

similar  character,  most  commonly  consisting  in  the  solution  of  the 
metal  composing  the  anode  and  in  the  deposition  of  another  metal 
or  of  hydrogen  on  the  cathode  (the  separation  of  free  hydrogen  being, 
however,  often  prevented  by  a  secondary  reaction  between  it  and  the 
electrolyte  or  the  electrode).  Thus,  in  the  Daniell  cell,  which  con- 
sists of  a  copper  electrode  in  a  copper  sulphate  solution  and  of  a  zinc 
electrode  in  a  zinc  sulphate  solution,  the  two  solutions  being  in  con- 
tact and  the  two  electrodes  connected  by  a  metallic  conductor,  the  zinc 
dissolves  and  the  copper  precipitates;  and,  in  the  Grove  cell,  con- 
sisting of  a  zinc  electrode  in  dilute  sulphuric  acid  and  a  platinum 
electrode  in  strong  nitric  acid,  zinc  dissolves  at  the  anode,  and  the 
hydrogen  primarily  produced  at  the  cathode  reduces  the  nitric  acid 
to  lower  oxides  of  nitrogen.  The  chemical  changes  involved  in  voltai" 
actions  do  not  differ,  therefore,  essentially  from  those  produced  by 
electrolysis. 

Pro&.  3.  State  what  chemical  change  takes  place  at  each  electrode 
when  electricity  passes  through :  a,  a  concentrated  solution  of  NaCl 
between  a  carbon  anode  and  an  iron  cathode ;  &,  a  solution  of  NaCl  be- 
tween a  silver  anode  and  a  metal  cathode  coated  with  AgCl;  c,  dilute 
H2SO4  between  a  lead  anode  and  a  cathode  of  lead  coated  with  PbO2; 
d,  a  solution  of  ZnSO4  between  a  zinc-amalgam  anode  and  a  cathode  of 
mercury  covered  with  Hg2SO4. 

43.  Faraday's  Law. — The  passage  of  electricity  through  an  elec- 
trolyte is  attended  at  each  electrode  by  a  chemical  change  involving 
a  number  of  chemical  equivalents  N  strictly  proportional  to  the  quan- 
tity of  electricity  Q  passed  through  and  dependent  on  that  alone. 
That  is :  Q  =  F  N,  where  F  is  a  constant  with  respect  to  all  varia- 
tions of  the  conditions,  such  as  temperature,  concentration,  current- 
strength,  current-density,  etc.  Such  variations  often  influence  the 
character  of  the  chemical  change,  but  not  the  total  number  of  equiva- 
lents involved.  The  law  is  applicable  to  concentrated,  as  well  as  to 
dilute  solutions,  and  to  fused  salts. 

The  constant  F  evidently  represents  the  quantity  of  electricity 
producing  a  chemical  change  involving  one  equivalent.  It  is  called 
a  faraday,  and  has  the  value  96500  coulombs.  One  coulomb  is  the 
quantity  of  electricity  flowing  per  second  when  the  current  is  one 
ampere. 

The  term  chemical  equivalent  in  the  above  statement  of  Faraday's 
law  signifies  the  oxidation  or  reduction  equivalent  of  the  substance, 


FARADAY'S  LAW  45 

in  the  sense  in  which  it  is  used  in  volumetric  analysis.  That  is,  one 
equivalent  of  any  substance  is  that  weight  of  it  which  is  capable  of 
oxidizing  one  atomic  weight  of  hydrogen,  or  which  has  the  same 
reducing  power  as  one  atomic  weight  of  hydrogen. 

Prob.  4.  Through  solutions  of  AgNO8,  AuCl3,  Hg2SO4,  and  HgClr 
placed  in  series,  9650  coulombs  are  passed.  How  many  grains  of  metal 
will  be  deposited  on  the  cathode  from  each  solution? 

Prob.  5.  A  Daniell  cell,  consisting  of  zinc  in  zinc  sulphate  solution 
and  copper  in  copper  sulphate  solution,  furnishes  a  current  of  0.1 
ampere  for  100  minutes.  How  many  grams  of  copper  deposit  and  of 
zinc  dissolve  in  the  cell? 

Pro  ft.  6.  How  long  must  a  current  of  5  amperes  be  passed  through 
dilute  sulphuric  acid  in  order  to  produce  at  27°  and  1  atm.,  a,  one  liter 
of  oxygen?  6,  one  liter  of  hydrogen? 

Prob.  7.  1930  coulombs  are  passed  through  a  solution  of  copper 
sulphate.  At  the  cathode  0.018  equivalent  of  copper  are  deposited. 
How  many  equivalents  of  hydrogen  are  set  free? 

44.  Relation  of  Faraday's  Law  to  the  Ionic  Theory. — Faraday's 
law  evidently  shows  that  electricity  is  transported  from  solution  to 
electrode,  or  in  the  reverse  direction,  only  by  the  constituents  of  the 
electrolyte;  and  that  one  equivalent  of  any  constituent  carries  the 
same  quantity  of  electricity,  namely,  one  faraday  or  96500  coulombs. 
The  dissociated  portions  of  these  constituents*  are  called  ions — the 
positively  charged  ones  cations,  and  the  negatively  charged  ones 
anions.  It  is  to  be  noted  that  the  flow  of  current  to  the  cathode  may 
be  brought  about  either  by  the  deposition  of  a  cation  on  the  electrode 
or  by  the  dissolving  from  the  electrode  of  a  substance  that  forms  an 
anion;  for  flow  of  negative  electricity  in  one  direction  produces  a 
current  having  the  same  electrical  effects  as  flow  of  positive  electricity 
in  the  other. 

Pro 6.  8.  When  one  faraday  is  passed  through  a  potassium  sul- 
phate solution,  not  only  one  equivalent  of  hydrogen,  but  also  one 
equivalent  of  potassium  hydroxide,  is  produced  at  the  cathode;  and 
not  only  one  equivalent  of  oxygen,  but  also  one  equivalent  of  sulphuric 
acid,  is  produced  at  the  anode.  Show  that  these  facts  would  not  be 
in  accord  with  Faraday's  law  if  the  ions  were  assumed  to  be  (K2O)  +  + 
and  (SO3)-~,  but  that  they  are  in  accord  with  it  if  the  ions  are  assumed 
to  be  K*  and  SO4-~. 

*The  two  ion-forming  constituents  themselves,  considered  without  reference 
to  the  extent  to  which  they  may  be  dissociated  from  each  other  in  the  solution, 
will  be  hereafter  called  the  ion-constituents.  Thus  the  Ion-constituents  of  potar 
slum  nitrate  are  K  and  NO3,  since  its  ions  are  K>  and  NOS-. 


46 


IONIC    PROPERTIES    OF    SOLUTIONS 


c 


M 


ELECTRICAL    TRANSFERENCE 

45.  Phenomenon  of  Electrical  Transference. — When  a  current  is 
passed  through  a  solution  of  a  salt,  base,  or  acid,  in  addition  to  the 
chemical  changes  taking  place  at  the  electrodes  in  accordance  with 
Faraday's  law,  a  certain  quantity  of  the  cation-constituent  is  trans- 
ferred from  the  neighborhood  of  the  anode  to  that  of  the  cathode,  and 
a  certain  quantity  of  the  anion-constituent  is 
transferred  in  the  reverse  direction.  This 
phenomenon  can  best  be  made  clear  by  the 
consideration  of  an  actual  transference  deter- 
mination. Consider,  for  example,  that  a  0.02 
normal  solution  of  sodium  sulphate  is  electro- 
lyzed  at  185  in  an  apparatus  like  that  shown 
in  Figure  5,  between  a  platinum  cathode 
(marked  — )  and  a  platinum  anode  (marked 
+).  To  avoid  stirring  of  the  solution,  the 
electrodes,  from  which  hydrogen  and  oxygen 
gases  are  evolved,  are  placed  near  the  surface; 
the  anode,  around  which  the  solution  becomes 
denser  during  the  electrolysis,  is  placed  near 
the  bottom  of  the  tube;  and  the  apparatus  is 
immersed  in  a  water-bath  kept  at  constant  tern-  FIGURE  5 

perature.  The  current  is  stopped  before  the  electrode-products  (the 
sodium  hydroxide  and  sulphuric  acid)  have  migrated  beyond  the  dotted 
lines  in  the  figure.  The  three  portions  of  the  solution  (called  the 
cathode-portion,  middle-portion,  and  anode-portion,  and  marked  0,  M, 
and  A,  respectively)  are  then  separately  removed  from  the  apparatus, 
and  submitted  to  analysis.  The  quantity  of  sodium  and  of  sulphate 
present  in  each  portion  is  compared  with  the  quantity  of  it  originally 
associated  with  the  weight  of  water  contained  in  the  portion.  It  is 
found,  if  the  experiment  has  been  successful,  that  the  middle- portion 
has  undergone  no  change  in  composition,  that  the  cathode-portion  has 
increased  its  sodium-content  and  decreased  its  sulphate-content,  and 
that  the  anode-portion  has  increased  its  sulphate-content  and  decreased 
its  sodium-content.  It  is  found,  per  faraday  of  electricity  passed 
through  the  solution,  that  the  sodium-content  has  increased  in  the 
cathode-portion  by  0.39  equivalent  and  has  decreased  in  the  anode- 


ELECTRICAL    TRANSFERENCE  47 

portion  by  the  same  amount,  and  that  the  sulphate-content  has  in- 
creased in  the  anode-portion  by  0.61  equivalent  and  has  decreased 
in  the  cathode-portion  by  the  same  amount. 

When  either  of  the  constituents  whose  transference  is  being  deter- 
mined is  deposited  on  or  dissolved  off  the  electrode,  as  is  the  case  when 
a  silver-nitrate  solution  is  electrolyzed  between  silver  electrodes,  the 
quantity  of  it  so  deposited  or  dissolved  must  evidently  be  determined 
either  by  direct  weighing  or  by  calculation  with  the  aid  of  Faraday's 
law,  and  be  subtracted  from  or  added  to  the  change  in  content  of  that 
constituent  in  the  electrode-portion. 

46.  Law  of  Transference. —  The  sum  of  the  number  of  equivalents 
of  the  cation  and  anion  constituents  (NC   and  NA)  transferred  in  the 
two  directions  is  equal  to  the  number  of  faradays  (N  )  passed  through 
the  solution.     That  is,   NC  +  NA  =  N.     This  equation  is  illustrated 
by  the  data  for  sodium  sulphate  given  above. 

In  the  case  of  mixtures  containing  various  ion-constituents 
(Cj,  C2,  . . .  AJ,  A2  . . .)  all  of  these  are  transferred,  and  the  expression 
of  the  law  of  transference  is :  NCj  +  NCs  . . .-  +  NAi  -f-  NA2  . . .  =  N. 

47.  Transference-Numbers. — The    equivalents    NC    of  cation-con- 
stituent transferred  are  in  general  not  equal  to  the  equivalents  NA   of 
anion-constituent  transferred.     The  experimentally  determined  ratios 
Nc/(Nc  +  NA)  and  NA/(NC  +  NA)  are  called  the  transference-numbers, 
TC  and  TA,  of  the  cation  and  anion,  respectively.    These  transference- 
numbers  evidently  represent  (since  xc  -}-  NA  =  N)  the  equivalents  of 
cation  or  anion-constituent  transferred  per  faraday.    Thus  in  the  case 
of  sodium  sulphate  TC  =  0.39  and  TA  =  0.61. 

Prob.  9.  Through  a  0.2  normal  solution  of  potassium  sulphate  be- 
tween platinum  electrodes  0.0075  faraday  is  passed  at  25°.  The  cathode- 
portion  after  the  electrolysis  was  found  to  contain  0.1450  g.  more  potas- 
sium than  was  originally  associated  with  the  weight  of  water  in  the 
portion.  What  is  the  transference-number  of  the  sulphate-ion? 

Prob.  10.  A  current  is  passed  at  25°  through  a  solution  of  16.64  g. 
Pb(NO3)2  in  1000  g.  water  between  lead  electrodes  until  0.1658  g.  silver 
is  deposited  in  a  coulometer  in  series  with  it.  The  anode  portion 
weighed  62.50  g.  and  yielded  on  analysis  1.123  g.  PbCrO4.  What  is  the 
transference-number  of  the  lead-ion?  Assume  that  lead  dissolves  off 
the  anode  in  accordance  with  Faraday's  law. 

Prob.  11.  A  current  of  0.1  ampere  is  passed  for  30  minutes  through 
a  0.2  normal  Ba(NOs),  solution  between  platinum  electrodes.  The 


48  IONIC    PROPERTIES    OF    SOLUTIONS 

transference-number  for  the  barium-ion  is  0.45.  What  changes,  expressed 
in  equivalents  of  the  ion-constituents,  result  in  the  anode-portion  and 
in  the  cathode-portion  from  the  electrolysis?  from  the  transference? 
What  is  the  net  result  of  these  two  effects  on  the  quantities  of  the 
various  compounds  present  in  each  portion? 

48.  Transference  in  Relation  to  the  Ionic  Theory.  The  Mechan- 
ism of  Conduction  in  Solutions. — Just  as  Faraday's  law  shows  that 
electricity  is  carried  from  the  solution  to  the  electrode  only  by  the  ion- 
constituents,  so  the  law  of  transference  (NC  -f-  NA  =  N)  shows  that 
through  the  solution  the  electricity  is  likewise  carried  only  by  the  ion- 
constituents.  The  only  difference  is  that  as  a  rule  only  one  kind  of 
ion-constituent  carries  the  electricity  to  the  electrode,  while  all  the 
ion-constituents  present  take  part  in  the  conduction  of  it  through 
the  solution. 

That  the  ion-constituents  do  move  through  the  solution  can  be 
shown  by  placing  the  solution  of  a  salt,  such  as  copper  sulphate  or 
potassium  permanganate,  whose  cation  or  anion  has  a  characteristic 
color,  beneath  a  solution  of  a  colorless  salt,  such  as  potassium  sulphate, 
and  applying  a  potential-difference  at  the  electrodes. 

The  movement  of  the  ion-constituents  is  explained  by  the  ionic 
theory  as  follows.  A  certain  fraction  of  the  molecules  of  a  salt  exists 
in  the  state  of  positively  and  negatively  charged  molecules,  called 
cations  and  anions.  When  a  solution  is  placed  between  electrodes 
that  are  at  different  potentials,  the  ions  in  virtue  of  their  charges 
are  subjected  to  an  electric  force  which  drives  them  through  the 
solution — the  cations  towards  the  cathode,  the  anions  towards  the 
anode;  while  the  unionized  molecules,  being  electrically  neutral,  are 
unaffected.  The  ions  are,  however,  constantly  uniting  to  form  union- 
ized molecules,  and  the  latter  are  constantly  dissociating  into  ions 
For  this  reason,  although  at  any  moment  only  the  ions  are  moving, 
the  resultant  effect  is  that  the  ion-constituent  as  a  whole  moves  con- 
tinuously towards  the  electrode.  The  rate  at  which  the  ion-constit- 
uent moves  is  equal  to  the  rate  at  which  the  ion  moves  multiplied 
by  the  ionization  of  the  salt ;  for  the  statement  that  a  certain  fraction 
of  the  molecules  is  ionized  is  equivalent  to  the  statement  that  any 
one  molecule  exists  in  the  form  of  its  ions  during  that  fraction  of 
the  time. 


ELECTRICAL    TRANSFERENCE  49 

49.  Transference  in  Relation  to  the  Mobility  of  the  Ions. — The 
electric  force  /  acting  on  any  charged  body  is  equal  to  its  charge  Q 
multiplied  by  the  potential-gradient;  that  is,  /  =  q(dE/dl),  where 
dE  is  the  change  of  potential  in  the  distance  dl.  Moreover,  the 
velocity  of  any  body  moving  through  a  medium  of  great  frictional 
resistance  is  proportional  to  the  force  acting  upon  it.  Therefore, 
since  the  resistance  to  the  motion  of  ions  through  solutions  is  very 
great,  the  velocity  u  of  any  given  ion  is  proportional  to  the  potential  - 
gradient;  that  is:  u  =  u(dE/dl),  where  u  is  the  velocity  under  unit 
potential-gradient,  called  the  mobility  of  the  ion.  The  velocity  of  the 
ion-constituent  is  evidently  also  proportional  to  the  potential-gradient. 

Prob.  12.  A  solution  containing  c  equivalents  of  a  salt  per  ccm. 
of  solution  is  placed  in  a  cylindrical  tube  of  cross-section  q  sqcm.  be- 
tween electrodes  I  cm.  apart,  at  which  a  potential-difference  E  is  applied. 
The  mobilities  of  the  cation  and  anion  constituents  in  tbis  solution  are 
uc  and  UA,  respectively.  Sketch  a  diagram  illustrating  these  conditions, 
and  derive  an  expression  for  the  number  of  equivalents  NC  and  NA  of  the 
cation  and  anion  constituents  which  migrate  through  any  cross-section 
of  the  solution  in  tbe  time  t. 

Since  the  migration  considered  in  Prob.  12  takes  place  also  through 
the  cross-sections  which  separate  the  middle  portion  from  the  two 
electrode  portions,  it  is  evident  that  the  equivalents  of  the  cation  and 
anion  constituents  transferred  are  to  each  other  as  their  mobilities. 
That  is,  NC  /NA  =  u  c  /  UA  ;  and  therefore : 

u 

—   and  T   = _ . 


Uc  +  UA  Uc  +  UA 

It  is  also  evident  that  the  transference-number  of  either  ion-constitu- 
ent is  the  fraction  of  the  current  which  is  carried  by  that  constituent. 

Prob.  18.  An  ordinary  transference  determination  is  made  at  18° 
with  0.1  normal  AgNO3  solution  in  a  cylindrical  tube  4  cm.  in  diameter 
between  silver  electrodes  30  cm.  apart.  Analysis  of  the  anode-portion 
shows  tbat  0.00207  equivalent  of  silver  have  migrated  out  of  it.  a.  Cal- 
culate the  distance  through  which  tbe  silver  migrated  during  the 
passage  of  the  current.  6.  The  potential-difference  applied  at  the  elec- 
trodes was  10  volts,  and  the  resulting  current  of  0.0395  ampere  was 
passed  for  3  hours.  Calculate  tbe  mobility  in  centimeters  per  second 
of  each  of  the  ion-constituents.  (Assume  that  tbe  concentration-changes 
at  the  electrodes  do  not  affect  the  potential -gradient.) 


C'A 


50  IONIC    PROPERTIES    OF    SOLUTIONS 

50.    The  Moving-Boundary  Method  of  Determining  Transference. 

— The  relation  derived  in  Prob.  12  forms  the  principle  of  another 
method  of  determining  transference.    In  this  method  the 
relative  rates  are  measured  at  which  the  two  boundaries 
of  a  solution  of  a  salt  CA  move  when  placed  between 
solutions  of  two  other  salts*  C'A  and  CA',  arranged  as 
in  Figure  6,  in  which  c  c  and  a  a  represent  the  original  c> 
positions  of  the  boundaries,  c'  c'  and  a'  of  their  positions 
after  a  certain  time.    It  is  evident  that  the  cation-con- 
stituent C  moves  the  distance  c  c'  while  the  anion-con-  a' 
stituent  A  moves  the  distance  a  of ;   and  that  they  are 
moving    under   the    same    potential-gradient,    since    they 
are  in  the  same  solution.    Therefore  the  ratio  c  c' /a  a'  is 
the  ratio  of  the  mobilities  UC/UA,  and  hence  of  the  transfer- 


, 


ence-numbers  TC/TA.  FIGURE  6 

The  boundaries  are  most  readily  seen  when  the  "indicator"  ions 
C'  and  A'  are  colored;  but  even  when  the  ions  are  all  colorless,  the 
boundaries  are  usually  visible  because  of  the  different  refractive  power 
of  the  adjoining  solutions. 

Prob.  14»  In  a  moving-boundary  experiment  an  apparatus  like  that 
represented  by  Figure  6  is  charged  at  18°  with  solutions  of  silver  nitrate 
at  the  bottom,  of  potassium  nitrate  in  the  middle,  and  of  potassium 
acetate  at  the  top.  The  lower  electrode,  which  is  of  silver,  is  made  the 
anode.  In  90  minutes  the  lower  boundary  moves  3.00  cm.  and  the  upper 
boundary  2.88  cm.  What  transference-numbers  can  be  derived  from 
these  facts,  and  what  are  their  values? 

51.    Change  of  Transference-Numbers  with  the  Concentration.  — 

The  transference-numbers  of  almost  all  uniunivalent  and  unibivalent 
salts  (except  the  halides  of  bivalent  metals)  remain  sensibly  constant 
as  the  concentration  increases,  so  long  as  it  does  not  exceed  a  moder- 
ate value,  say  0.1  normal.  For  example,  the  values  at  18°  of  the 
sodium  transference-number  TNa  in  sodium  chloride  solutions  at  vari- 
ous concentrations  c  are  as  follows  : 

TNa    .......     0.396        0.396        0.395        0.393        0.388        0.369 

o   .........    0.005        0.020        0.050        0.100        0.300        1.000  normal. 

*In  order  that  the  boundaries  remain  sharp  it  is  evidently  necessary  that 
the  ion  C'  have  a  smaller  velocity  than  the  ion  C,  and  that  the  ion  A'  have  a 
smaller  velocity  than  the  ion  A  ;  for  otherwise  the  ions  C'  and  A'  would  enter  the 
solution  of  the  salt  CA,  producing  a  mixture  at  each  boundary. 


ELECTRICAL    TRANSFERENCE  51 

The  constancy  of  these  numbers  in  the  dilute  solutions  shows  that 
the  ratio  of  the  velocities  of  the  two  ion-constituents  does  not  vary, 
which  might  be  expected  to  be  true  so  long  as  they  are  moving  through 
a  medium  which  is  substantially  the  same  as  pure  water.  At  higher 
concentrations  the  transference-number  often  changes  rapidly  with 
increasing  concentration.  This  may  be  due  to  a  variety  of  causes. 
Thus  it  may  arise  from  a  change  in  the  frictional  resistance  of  the 
medium;  from  hydration  of  the  ions,  which  causes  water  to  be  trans- 
ferred and  thus  affects  the  transference  value,  since  this  is  computed 
under  the  assumption  that  the  water  is  stationary;  from  existence  of 
complex  ions,  which  in  concentrated  solutions  are  more  likely  to  be 
present  in  considerable  quantity. 

52.  Determination  of  the  Composition  of  Ions  by  Transference 
Experiments. 

Proo.  15.  When  one  faraday  is  passed  through  a  solution  of  potas- 
sium silver  cyanide  (KCN.AgCN)  the  cathode-portion  loses  1.40  equiva- 
lents of  silver  and  0.80  equivalent  of  cyanogen,  and  gains  0.60  equivalent 
of  potassium.  Explain  what  this  shows  in  regard  to  the  composition  of 
the  ions  and  their  transference-numbers. 

Proo.  16.  A  transference  experiment  is  made  by  passing  0.01  fara- 
day through  a  solution  0.2  formal  in  AgNO3  and  0.6  formal  in  NH8 
between  silver  electrodes  at  18°.  The  anode-portion  is  found  to  gain 
0.0053  equivalent  of  silver  and  to  lose  0.0094  formula-weight  NH3.  Ex- 
plain what  this  shows  in  regard  to  the  composition  of  the  ions.  (Silver 
dissolves  at  the  anode  in  accordance  with  Faraday's  law.) 

Proo.  17. — Determination  of  the  Hydration  of  Ions. — In  a  transfer- 
ence experiment  0.0525  faraday  was  passed  at  25°  through  a  solution 
placed  between  silver  electrodes  and  containing  1.12  formula-weights  of 
NaCl  and  0.073  formula-weights  of  raffinose  (C^H^O^)  in  1000  g.  of 
water.  The  anode-portion  was  found  by  analysis  to  contain  0.72  g.  less 
water  and  1.115  g.  less  NaCl  than  was  originally  associated  with  the 
raffinose  present  in  that  portion,  a.  Calculate  the  number  of  mols  of 
water  and  the  number  of  equivalents  of  sodium  transferred  per  faraday 
from  the  anode  to  the  cathode,  assuming  that  the  raffinose  does  not 
migrate,  o.  Assuming  no  hydration  of  the  chloride  ion,  calculate  the 
number  of  molecules  of  water  associated  with  the  atom  of  sodium  in 
the  sodium  ion.  c.  Assuming  the  chloride  ion  is  hydrated  with  x  mole- 
cules of  water,  calculate  the  hydration  of  the  sodium  ion. 

Note. — Similar  experiments  made  with  other  halides  have  given  the 
following  values  for  the  number  of  molecules  of  water  contained  in 
other  ions,  assuming  the  number  in  the  chloride  ion  to  be  x  molecules : 
H+    :  0.28  -f  0.185#.  K+    :  1.3  -f-  1.02#. 

Cs+   :  0.67  -f  1.03a?.  U+   :  4.7  -f-  2.29#. 


52  IONIC    PROPERTIES    OF    SOLUTIONS 


ELECTRICAL    CONDUCTANCE 

53.  Conductance,  Specific  Conductance,  and  Equivalent  Conduct- 
ance.— According  to  Ohm's  law,  the  current  I  flowing  between  two 
points  of  a  conductor  is  proportional  to  the  potential-difference  E  at 
those  points.  The  ratio  of  the  current  to  the  potential-difference  is 
called  the  conductance  L;  and  the  inverse  ratio,  the  resistance  R. 

That  is, 

I/E  =  L,  and  E/I  =  R. 

When   the   current   is   expressed   in   ani^eres   and   the   potential 
difference  in  volts,  the  resistance  is  in  ohms  and  the  conductance  in 
reciprocal  ohms.    An  ohm  is  the  resistance,  and  a  reciprocal  ohm  the 
conductance,  of  a  column  of  mercury  at  0°   one  square  millimeter 
in  cross-section  and  106.3  centimeters  long. 

The  conductance  of  a  homogeneous  body  of  uniform  cross-section 
is  proportional  to  its  cross-section  q  and  inversely  proportional  to  its 
length  Z.  That  is,  L  =  L  q/l.  The  proportionality-factor  L",  which 
is  the  conductance  when  the  cross-section  is  one  square  centimeter 
and  the  length  one  centimeter,  is  called  the  specific  conductance.  Its 
reciprocal  is  called  the  specific  resistance. 

In  the  case  of  a  salt  in  solution,  the  term  equivalent  conductance 
A  is  employed  to  denote  the  conductance  of  that  volume  of  solution 
which  contains  one  equivalent  of  salt,  when  placed  between  parallel 
electrodes  one  centimeter  apart.  Thus  the  equivalent  conductance 
Ap.i  of  a  salt  in  0.1  normal  solution  represents  the  conductance  of 
10,000  cubic  centimeters  of  that  solution  when  placed  between  paral- 
lel electrodes  one  centimeter  apart. 

The  equivalent  conductance  of  dissolved  salts  increases  with 
decreasing  concentration  at  first  rapidly,  then  more  slowly,  and 
approaches  a  maximum  value  as  the  concentration  approaches  zero. 
This  is  illustrated  by  the  following  data  at  18° : 

Fqniv.  per  liter....         1  0.1         0.01        0.001          0.0001  0.0 

A  for  NaCl 74.4      92.0      102.0        106.5          108.1  109.0 

A  for  K2S04 71.6       94.9       115.8         126.9  130.8  133.0 

The  equivalent  conductance  of  largely  ionized  solutes  increases 
with  rising  temperature.  At  18°  the  increase  in  dilute  solution  is 
from  2.1  to  2.5%  per  degree  in  the  case  of  salts,  and  about  1.6%  per 
degree  in  the  case  of  strong  acids. 


. 

ELECTRICAL    CONDUCTANCE  53 

Prol.  18.  What  is  the  specific  conductance  in  reciprocal  ohms  of 
mercury  at  0°  ? 

Prob.  19.  a.  Derive  the  algebraic  relation  expressing  equivalent  con- 
ductance in  terms  of  specific  conductance  and  concentration  in  equiva- 
lents per  cubic  centimeter.  6.  Calculate  the  specific  conductance  of  the 
sodium  chloride  solutions  whose  equivalent  conductances  are  given 
above. 

Profc.  20.  A  0.1  normal  solution  of  silver  nitrate  at  18°  is  placed 
in  a  tube  4  cm.  in  diameter  between  silver  electrodes  12  cm.  apart.  A 
potential  difference  of  20  volts  at  the  electrodes  produces  a  current  of 
0.1976  ampere.  Calculate  the  conductance,  the  specific  conductance, 
and  the  equivalent  conductance  of  the  solution. 

Prob.  21.  The  equivalent  conductance  of  a  0.01  normal  CuSO4  solu- 
tion at  18°  is  71.7  reciprocal  ohms.  What  is  the  resistance  of  a  column 
of  it  20  cm.  long  and  5  sqcm.  in  cross-section? 

54.  Relation  of  Conductance  to  the  Mobility  of  the  Ion-Constit- 
uents. 

Pro 6.  22.  In  0.2  normal  NaNO8  solution  the  mobility  at  18°  of  the 
sodium  is  0.000505  and  that  of  the  nitrate  0.000356  cm.  per  second. 
Calculate  the  conductance  of  a  column  of  the  solution  10  cm.  long  and 
2  sqcm.  in  cross-section.  Note  that  the  conductance  of  any  conductor 
is  the  quantity  of  electricity  in  coulombs  which  passes  per  second  when 
the  potential-difference  at  the  ends  is  1  volt;  and  that  the  conduct- 
ance of  a  salt  solution  is  the  sum  of  the  conductances  of  its  two  ion- 
constituents. 

Pro 6.  23.  Show  that  the  specific  conductance  17  and  equivalent  con- 
ductance A  are  expressed  by  the  equations : 

L  =  LC  +  LA  =  (UC  +  UA)  P  C,    and   A  =  Ac  +  AA  =  (uc  +  UA)  F, 
where  c  is  the  concentration  in  equivalents  per  cubic  centimeter  of  a 
solution  in  which  the  mobilities  of  the  ion-constituents  are  uc  and  UA . 

55.  Relation    of    Equivalent    Conductance    to    lonization. — The 

equation  A  =  (uc  -f-  UA)  F  shows  that  the  observed  increase  of 
equivalent  conductance  with  decreasing  concentration  is  due  to  an 
increase  in  the  mobilities  of  the  ion-constituents.  In  the  case  of  dilute 
solutions  of  di-ionic  substances  this  increase  in  mobility  doubtless 
arises,  in  the  way  described  in  Art.  48,  from  a  proportionate  increase 
in  the  ionization  of  the  substance;  for  up  to  moderate  concentrations 
the  resistance  to  the  movement  of  the  ions  is  presumably  the  same  as 
in  pure  water.  The  maximum  value  attained  at  zero  concentration 
evidently  corresponds  to  complete  ionization.  The  ratio  of  the  equiva- 
lent conductance  A  at  any,  not  too  high,  concentration  to  the  equivalent 


54  IONIC  PROPERTIES  OF  SOLUTIONS 

conductance  AO  at  zero  concentration  is  therefore  equal  to  the  ioniza- 
tion  7  in  the  case  of  a  di-ionic  substance.  That  is,  A  /  AO  —  7.  The 
A  „- value  is  ordinarily  obtained  by  extrapolation.  When  the  substance 
is  completely  ionized  the  mobilities  uc  and  UA  and  equivalent  con- 
ductances AC  and  AA  of  the  two  ion-constituents  become  identical 
with  those,  uc+,  UA-,  Ac+,  AA-,  of  the  corresponding  ions.  Therefore 
A0  =  AC+  -f-  AA-  ==  (uc+  -j-  UA-)  F-  These  ion-mobilities  and  ion- 
conductances  have  at  each  temperature  a  certain  value  for  each  ion, 
whatever  be  the  other  ion  in  the  solution. 

Proo.  24.  In  0.1  normal  solution  at  18°,  HC1  and  HC2H8O2  are 
92%  and  1.35%  ionized,  respectively.  With  what  velocity  in  centimeters 
per  hour  does  the  hydrogen  migrate  through  each  solution  when  the 
potential-gradient  is  10  volts  per  centimeter?  The  mobility  of  the 
hydrogen-ion  at  18°  is  0.00326  cm.  per  second. 

Pro  6.  25.  At  18°  the  equivalent  conductance  of  0.1  normal  NH4OH 
is  3.1  reciprocal  ohms.  The  equivalent  conductance  at  zero  concentra- 
tion for  NH4C1,  KC1,  and  KOH  is  130.2,  130.0,  and  239  reciprocal  ohms, 
respectively.  What  is  the  ionization  of  ammonium  hydroxide  in  0.1 
normal  solution? 

The  determination  of  the  ionization  of  polyionic  substances  is 
complicated,  even  in  dilute  solutions,  by  the  fact  that  these  substances 
dissociate  in  two  ways;  thus  sulphuric  acid  dissociates  partially  into 
H+  and  HSO4~,  and  partially  into  H+,  H+,  and  SO4".  The  conductance- 
ratio  for  such  substances  has  therefore  no  simple  significance. 

In  even  moderately  concentrated  solutions  the  frictional  resistance 
to  the  motion  of  the  ions  must  be  appreciably  different  from  that  in 
pure  water ;  and  the  conductance-ratio  A  /  A0  cannot  therefore  be  an 
exact  measure  of  the  ionization.  An  approximate  estimate  of  the 
variation  of  this  frictional  resistance  is  given  by  the  ratio  of  the  vis- 
cosity 77  of  the  solution  to  the  viscosity  77  0  of  pure  water  at  the  same 
temperature.  This  viscosity-ratio  77/770  is  determined  by  measuring 
the  relative  times  required  for  equal  volumes  of  the  solution  and  of 
pure  water  to  flow  through  the  same  capillary  tube,  when  subjected 
to  the  same  pressure.  A  film  of  the  liquid  adheres  firmly  to  the  walls 
of  the  tube ;  and  the  phenomenon  of  the  flow  consists  essentially  in  the 
slipping  of  the  successive  cylindrical  shells  of  liquid  past  one  another. 
Viscosity  is  therefore  a  property  which  depends  on  the  frictional  re- 
sistance to  the  motion  of  the  molecules  of  the  liquid  past  one  another ; 
and  it  may  be  expected  to  be  roughly  proportional  to  the  frictional 


ELECTRICAL    CONDUCTANCE 


55 


resistance  to  the  motion  of  ions  through  the  same  liquid.  By  taking 
the  viscosity  into  account  an  ionization-value  can  be  derived  which  is 
more  accurate  than  that  given  by  the  conductance-ratio  alone. 

Pro&.  26.  a.  At  18°  the  viscosities  of  0.1-normal  and  1.0-normal  NaCl 
solutions  are  respectively  1.009  and  1.086  times  as  great  as  that  of  pure 
water.  With  the  aid  of  these  data  and  those  given  in  Art.  53  calculate 
the  ionization*of  NaCl  at  18°  at  these  two  concentrations.  6.  Formulate 
an  algebraic  expression  for  the  ionization  in  terms  of  the  conductances 
and  viscosities. 

Note. — The^  viscosity-ratios  ?}/TIO  for  some  other  salts  at  18°  at 
1-normal  are  as  follows : 

Salt  KC1          LiCl        MgCl2      K2SO4      MgSO4 

?A0         0.982         1.150        1.213        1.101         1.381 
The  deviation  of  the  ratio  from  unity  at  0.1-normal  is  approximately 
one-tenth  of  that  at  1-normal. 

56.  Mobility  and  Conductance  of  the  Separate  Ions. — Although 
only  the  sum  of  two  ion-conductances  is  given  by  the  A0- value,  the 
conductances  of  the  separate  ions  may  be  obtained  by  combining  it 
with  the  transference-number. 

Pro 6.  27.  a.  Calculate  from  the  conductance  and  transference  data 
given  in  Arts.  51  and  53  for  sodium  chloride  the  equivalent  conduct- 
ances of  sodium-ion  and  chloride-ion.  6.  At  18°  the  value  of  A0  for 
KC1  is  130.0,  for  KNO3  is  126.3,  and  for  HNO3  is  377  reciprocal  ohms. 
From  these  data  and  the  result  obtained  in  a,  calculate  a  value  for 
the  equivalent  conductance  of  each  of  the  ions  of  these  salts,  c.  Cal- 
culate the  cation-transference-number  at  18°  for  nitric  acid  in  dilute 
solution.  (The  experimentally  found  value  is  0.839  at  0.005  normal.) 

The  following  table  contains  the  values  of  the  equivalent  conduct- 
ance A  of  some  important  ions  at  18°,  and  the  values  of  its  fractional 
increase  <x  (equal  to  (efA/A  )/dT)  with  the  temperature  at  18°. 

^  A  ex  A  oc 

H+  315  0.0154  OH-  174  0.0180 

Li*  33.3  265  F-  46.7  238 

Na+  43.4  244  Cl~  65.5  216 

K>  64.5  217  Br-  67.7  215 

NH4+  64.7  222  I-  66.6  213 

Ag+  54.0  229  NO3-  61.8  205 

Mg++  45.9  256  C1O3-  55.1  215 

Ca++  51.9  247  BrO3-  47.6  216 

Ba++  55.4  238  TO,-  34.0  234 

Pb++  60.8  243  C2H3O2-  35.  238 

Cu++  45.9  254  SO4"  68.5  227 

Zn++  47.0  254  C2<X"  63.0  220 


56  IONIC    PROPERTIES    OF   SOLUTIONS 

Prol).  28.  a.  From  a  consideration  of  the  table  given  in  the  preceding 
text  state  the  relation  that  exists  between  the  ion-conductances  and  the 
atomic  weights  of  the  different  elementary  ions  belonging  to  the  alkali 
group  and  to  the  alkaline-earth  group.  b.  Suggest  an  explanation  of  this 
apparently  anomalous  phenomenon,  c.  The  viscosity  of  water  at  18° 
decreases  2.6%  per  degree.  State  and  explain  any  relation  that  this  value 
has  to  the  temperature-coefficients  of  the  ion-conductances. 

From  these  ion-conductances  the  A0-value  for  the  various  salt? 
can  be  obtained  by  simple  addition.  This  fact  is  of  especial  impor- 
tance in  the  case  of  substances  for  which  the  A0-value  cannot  be 
obtained  from  conductance  measurements  by  extrapolation.  This  is 
true  of  weak  bases  and  acids,  such  as  ammonium  hydroxide  and  ucetif 
acid,  whose  ionization  is  far  from  complete  even  in  dilute  solution, 
and  of  salts,  such  as  ammonium  acetate,  which  are  appreciably 
hydrolyzed  in  dilute  solution. 

From  the  ion-conductances  can  be  calculated  also  the  specific  con- 
ductance of  any  solution  in  which  the  ion-concentrations  are  known. 

Prol).  29.  In  a  solution  at  18°  containing  0.10  equivalent  NaCl  and 
0.05  equivalent  KC1,  each  of  the  salts  is  82%  ionized.  Calculate  the  ion- 
concentrations,  and  from  them  the  specific  conductance  of  the  solution. 

57.    Ion-Concentrations  Derived  from  Conductance  Measurements. 

Prob.  30. — Determination  of  the  Solubility  of  Slightly  Soluble  Sub- 
stances.— When  water  at  18°  is  saturated  with  silver  chloride,  its  specific 
conductance  increases  by  1.25  X  10~8  reciprocal  ohm.  Assuming  that  the 
silver  chloride  is  completely  ionized,  find  its  solubility  in  equivalents 
per  liter. 

Prob.  81.  At  25°  the  specific  conductance  of  pure  water  due  to  its 
ionization  into  H+  and  OH-  is  0.055  x  10-'  reciprocal  ohm.  What  is  the 
concentration  of  these  ions  in  equivalents  per  liter?  (In  calculating 
the  ion-conductances  at  25°,  use  the  temperature-coefficients  given  in  the 
above  table.) 


IONIZATION  OF  SUBSTANCES  57 

THE  IONIZATION  OF  SUBSTANCES  OF  DIFFERENT  TYPES 

58.  lonization  of  Substances  as  Derived  from  the  Conductance- 
Ratio.  —  Almost  all  salts  (but  not  acids  or  bases)  of  the  same  valence 
type  (i.  e.,  those  whose  ions  have  the  same  electric  charge  or  valence) 
have  at  the  same  concentration  and  temperature  not  far  from  the 
same  value  of  the  ratio  A>?  /  A8?0.  The  average  values  of  this  quantity 
at  18°  for  the  three  simplest  types  of  salts  are  as  follows  : 

Type  Example        0.001          0.01  0.02  0.05  0.1  normal 

Uniunivalent  KNO3        0.98        0.93        0.91        0.87        0.84 


Unibivalent  KO*        a95        °'87        °*84        °'78        °*73 

Bibivalent  MgSO4      0.86        0.64        0.56        0.47        0.41 

Few  salts  of  the  uniunivalent  or  unibivalent  type  have  ratios  at 
0.1  normal  differing  by  more  than  five  per  cent,  from  the  average 
values  given  above.  There  are,  however,  certain  salts  of  the  unibiva- 
lent type  which  form  marked  exceptions  to  the  rule  ;  thus  at  0.1  normal 
the  ratio  for  cadmium  chloride  is  45%,  and  the  ratios  for  the  three 
mercuric  halides  are  all  less  than  0.1%.  The  value  of  the  ratio  for  all 
the  types  of  salts  decreases  with  rising  temperature,  but  only  to  a 
slight  extent. 

It  has  already  been  stated  in  Art.  55  that  the  ratio  A^/A0^0  is  ap- 
proximately equal  to  the  ionization  up  to  fairly  large  concentrations 
in  the  case  of  uniunivalent  substances.  In  the  case  of  unibivalent 
substances  this  is  not  true,  owing  to  the  fact  that  they  ionize  partially 
in  two  ways  —  with  formation  of  the  intermediate  ion  as  well  as  with 
formation  of  the  ultimate  ions  to  which  the  A0-value  corresponds.  In 
the  case  of  salts  of  this  unibivalent  type  the  proportion  of  the  inter- 
mediate ion  present  is  not  accurately  known;  up  to  concentrations  of 
0.1-0.2  normal  it  is,  however,  so  small  that  only  a  small  error  results 
from  neglecting  it  in  calculating  from  the  ratio  A?  /  AOT?O  the  concen- 
tration of  the  ultimate  univalent  ion;  but  a  much  larger  error  results 
in  thus  calculating  the  concentration  of  the  bivalent  ion. 

Prob.  82.  At  18°  the  specific  conductance  of  a  0.05  formal  K2SO4 
solution  at  18°  is  0.00949.  a.  Formulate  an  expression  for  this  specific 
conductance  in  terms  of  the  equivalent  concentrations  and  conductances 
of  the  three  kinds  of  ions  that  may  be  present.  6.  Calculate  with  the  aid 
of  this  expression  the  equivalent  concentrations  of  K+  and  of  SO4", 
assuming  that  these  are  the  only  ions  present,  c.  Calculate  these  equiva- 
lent concentrations,  assuming  that  20%  of  the  salt  is  ionized  into  KSO4- 


58  IONIC    PROPERTIES    OF    SOLUTIONS 

and  K+,  and  that  the  equivalent  conductance  of  KSO4-  is  35.  d.  Tabulate 
for  cases  &  and  c  the  fraction  of  the  total  sulphate  which  exists  as  SO4~-, 
KSO4-,  and  K2SO4,  respectively. 

Acids  and  bases,  unlike  salts,  exhibit  at  any  moderate  concentra- 
tion, such  as  0.1  normal,  every  possible  degree  of  ionization  between 
a  small  fraction  of  one  per  cent,  and  90  to  95%.  There  is,  to  be  sure, 
a  fairly  large  group  of  monobasic  acids  and  monacidic  bases,  includ- 
ing HC1,  HBr,  HI,  HNO3,  HC1O3,  KOII,  NaOH,  LiOH,  which  have 
ionization-values  comparable  with  those  of  the  uniunivaler.it  salts. 
with  which  therefore  they  may  be  classed.  But  outside  of  thia  group 
all  possible  values  are  met  with,  as  illustrated  by  the  following  values 
of  the  percentage  ionization  (1007)  at  25°  and  0.1  formal:  H2SO3, 
34%  (into  H+.  and  HSCV)  J  H3PO4,  28%  (into  H+  and  H2P(V) ; 
HN02,  7%;  HC2H302,  1.3%;  H2CO3,  H2S,  HC1O,  HCN,  HB02,  all 
less  than  0.2%. 

Polybasic  acids  are  known  to  ionize  in  stages,  giving  rise  to  the 
intermediate  ion;  and  the  first  hydrogen  is  almost  always  much  more 
dissociated  than  the  second,  and  the  second  much  more  than  the  third. 
Thus  H,SO3  at  0.1  normal  at  25°  is  about  34%  dissociated  into  IT 
and  HSO3",  and  less  than  0.1%  dissociated  into  H+  and  SO8".  Meth- 
ods by  which  the  dissociation  of  the  successive  hydrogens  can  be 
determined  will  be  referred  to  in  Art.  84. 

59.  Comparison  of  Ionization- Values  Derived  from  the  Conduct- 
ance-Ratio and  from  the  Freezing-Point-Lowering. — It  has  been  shown 
that  the  ionization  of  substances  can  be  determined  with  the  aid  of 
two  entirely  distinct  principles.  One  of  these,  based  on  the  increase 
in  the  number  of  mols  produced  by  ionization,  involves  the  assump- 
tion that  the  ions  and  unionized  substance  have  the  normal  effect  on 
the  vapor-pressure  and  related  properties  of  the  solvent.  The  other 
principle,  according  to  which  the  ionization  is  equal  to  the  conduct- 
ance-ratio, assumes  that  the  decrease  in  the  mobility  of  the  ion- 
constituents  with  increasing  concentration  is  due  solely  to  the  decrease 
in  ionization  of  the  substance.  A  comparison  shows  that  the  ioniza- 
tion-values derived  in  these  two  ways  agree  with  each  other  within 
2  or  3%  in  the  case  of  most  uniunivalent  substances  up  to  0.1  normal. 

The  comparison  of  the  ionization-values  obtained  by  the  two 
methods  in  the  case  of  unibivalent  salts  is  of  complicated  significance 
because  of  the  presence  of  the  intermediate-ion. 


REVIEW  PROBLEMS  59 

Pro 6.  S3  a.  Calculate  the  number  of  mols  per  formula-weight  of 
K2SO4  in  a  0.05  formal  solution  of  it  at  18°  from  the  results  obtained  in 
Prob.  32  (under  the  two  assumptions  that  the  salt  ionizes  without 
formation  of  the  intermediate  ion  and  that  it  ionizes  with  formation  of 
20%  of  that  ion).  b.  State  how  the  number  of  mols  per  formula-weight 
of  salt  actually  present  in  this  K2SO4  solution  could  be  experimentally 
determined. 


REVIEW  OP  THE  IONIC  PROPERTIES  OF  SOLUTIONS 

60.    Review  Problems. 

Prob.  34'  Give  an  expression  for  the  transference-number  in  terms 
(a)  of  the  number  of  equivalents  transferred;  (o)  of  the  mobilities  of 
the  ion-constituents;  (c)  of  the  conductances  of  the  ions. 

Prob.  35.  Express  algebraically  the  relations  (a)  between  resistance, 
conductance,  specific  conductance,  and  equivalent  conductance;  (6)  be- 
tween specific  conductance  and  ion-mobilities. 

Prob.  36.  Calculate  the  number  of  equivalents  of  sodium  that  would 
be  transferred  per  faraday  from  the  anode  to  the  cathode  portion  if  the 
solution  named  in  Prob.  30  were  electrolyzed. 

Pro  6.  57.  What  changes  in  the  hydrogen,  chlorine,  and  platinum 
content,  expressed  in  terms  of  the  number  of  atomic  weights  of  each 
element,  take  place  in  the  cathode  portion  when  4825  coulombs  are  passed 
through  a  solution  of  chlorplatinic  acid  (HaPtCl6)  at  25°?  Assume  that 
this  acid  dissociates  only  into  H+  and  PtCl6~-  ions,  whose  equivalent 
conductances  at  25°  are  350  and  68,  respectively.  The  cathode  process 
consists  only  in  the  deposition  of  platinum. 

Prob.  38.  Sulphuric  acid  in  0.05  formal  solution  at  25°  consists  of 
6%  H2SO4,  67%  HSO4-,  27%  SO4=,  and  the  corresponding  amount  of  H*. 
The  equivalent  conductances  at  25°  of  H*,  HSO4-,  and  SO4"  are  350. 
35,  and  80,  respectively,  a.  Calculate  the  specific  conductance  of  the 
solution  at  25°.  b.  Calculate  the  change  in  the  equivalents  of  hydrogen 
in  the  cathode  portion  when  one  faraday  is  passed.  (Note  that  hydro- 
gen is  transferred  both  in  the  form  of  H+  and  of  HSO4~  ions.) 


CHAPTER   V 
THE  RATE  OF  CHEMICAL  REACTIONS 


THE    EFFECT    OF    CONCENTRATION 

61.  Definition  of  Reaction-Kate. — The  rate  of  a  chemical  reaction 
between  gaseous  or  dissolved  substances  may  be  defined  to   be  the 
number  of  equivalents  of  each  of  the  reaction -products  produced  per 
liter  in  an  infinitesimal  time  divided  by  that  time ;  that  is,  the  reaction- 
rate  is  dc/dtf  where  dc  signifies  the  increase  of  the  equivalent  concen- 
tration of  each  of  the  reaction-products  in  the  time  dt.    As  to  the 
significance  of  the  terms  equivalent  and  equivalent  concentration  see 
Arts.  4  and  27.    It  is  in  some  cases  simpler,  especially  in  dealing  with 
reactions  between  gases  or  with  reactions  that  take  place  in  stages,  to 
employ  molal  concentration  (c)  in  place  of  equivalent  concentration 
(c) ;  but  when  this  is  done  it  will  be  so  specified. 

Profc.  1.  Sodium  hydroxide  and  methyl  acetate  in  dilute  solution 
react  with  each  other  (forming  sodium  acetate  and  methyl  alcohol)  at 
a  rate  which  at  any  moment  is  proportional  to  the  concentration  of  the 
sodium  hydroxide  and  to  the  concentration  of  the  methyl  acetate  at 
that  moment.  At  25°  in  a  solution  0.01  normal  in  each  of  these  sub- 
stances the  rate  at  which  sodium  acetate  is  produced  is  at  the  start 
0.00118  equivalents  per  liter  per  minute.  What  will  be  the  concentra- 
tion of  the  sodium  acetate  after  10  minutes? 

62.  The  Law  of  Concentration-Effect. — The  rate  of  any  chemical 
reaction  which  takes  place  between  perfect  gases  or  perfect  solute? 
at   a   constant   temperature   completely   in   one   direction   is   propor- 
tional to  the  concentration  of  each  of  the  reacting  substances  at  the 
moment  in  question. 

This  law  is  exact  only  for  perfect  gases  or  perfect  solutions;  but, 
like  the  other  limiting  laws,  it  holds  true  approximately  up  to  moder- 
ate concentrations.  The  statement  of  the  law  here  given  is  a  pro- 
visional one  which  will  be  made  more  complete  after  the  mechanism 
of  reactions  has  been  considered. 


EFFECT  OF  CONCENTRATION  61 

Pro b.  2.  Express  this  law  in  the  form  of  a  differential  equation 
for  the  general  case  that  any  number  of  substances  A,  B,  C,  .  .  .  react 
with  one  another.  Represent  by  COA,  COB>  C0c.  •  '  the  concentrations 
of  the  reacting  substances  at  time  zero,  and  by  c  the  concentration  of 
the  reaction-products  which  has  resulted  at  any  time  t. 

The  proportionality-constant  which  occurs  in  the  equation  ex- 
pressing the  law  of  concentration-effect  is  called  the  specific  reaction- 
rate  (&).  It  evidently  represents  the  rate  which  the  reaction  would 
have,  under  the  assumption  of  proportionality,  if  the  concentrations 
of  all  the  reacting  substances  were  unity  (one  equivalent  per  liter). 

63.  Eeactions  of  the  First  Order. — The  expression  for  the  reaction- 
rate  is  simplest  in  the  case  of  reactions  in  which  only  one  substance 
undergoes  a  change  in  concentration.    Such  reactions,  whose  rate  is 
expressed  by  a  differential  equation  of  the  first  degree,   are  called 
reactions  of  the  first  order. 

Pro b.  3.  Cane-sugar  in  a  dilute  aqueous  solution  containing  hydro- 
chloric acid  undergoes  hydrolysis  according  to  the  reaction: 

C.A.OU  +  H20  =  C^O,  (glucose)  +  C6HUO6(  fructose). 
Formulate  the  differential  equation  expressing  the  rate  at  which  the 
hydrolysis  takes  place. 

Note. — In  this  case  the  concentration  of  the  water  does  not  change 
appreciably,  since  it  is  present  in  such  large  excess ;  and  the  concentra- 
tion of  the  acid  does  not  change  at  all,  since  it  acts  catalytically,  accel- 
erating the  reaction  without  being  consumed  by  it.  It  is  not  usual  in 
formulating  the  reaction-rate  equation  or  in  evaluating  the  specific 
reaction-rate  to  take  into  account  the  concentrations  of  substances 
which,  like  these  just  mentioned,  do  not  change  with  the  progress  of  the 
reaction. 

Prob.  4.  Integrate  the  equation  obtained  in  Prob.  3  between  the 
time  limits  t  =  0  and  t  =  t  and  the  corresponding  concentration  limits. 

Prob.  b.  In  a  solution  at  48°  containing  0.3  mol  of  cane-sugar  in  a 
liter  of  0.1  normal  HC1,  it  is  found  (by  measuring  with  a  polarimeter 
the  change  in  the  optical  rotatory  power)  that  32%  of  the  sugar  is 
hydrolyzed  in  20  minutes,  a.  Calculate  the  specific  reaction-rate  and 
the  actual  rates  at  the  beginning  and  at  the  expiration  of  20  minutes. 
b.  Calculate  the  percentage  of  sugar  that  will  be  hydrolyzed  in  40  min- 
utes, c.  Calculate  the  percentage  of  sugar  that  would  be  hydrolyzed 
in  20  minutes,  if  0.3  mol  were  dissolved  in  10  liters  (instead  of  1  liter) 
of  0.1  normal  HC1. 

64.  Reactions  of  the  Second  and  Third  Orders. 

Prob.  6.  a.  Formulate  the  differential  equation  expressing  the  rate 
in  dilute  solution  of  the  reaction  between  two  suhstances  A  and  B. 


62  RATE  OF  CHEMICAL  REACTIONS 

b.  Integrate  this  equation  between  the  time  limits  t  =  0  and  t  =  t 
for  the  case  that  the  initial  concentrations  of  the  two  substances  have 
the  same  value  c0  Integrate  the  equation  (with  the  aid  of  a  table  of 
integrals,  if  preferred)  also  for  the  cas0  that  the  initial  concentrations 
COA  and  COB  are  different  from  each  other. 

A  reaction  whose  rate  is  expressed  by  the  differential  equation 
formulated  in  Prob.  6  is  called  a  reaction  of  the  second  order.  And 
in  general,  the  order  of  a  reaction  is  said  to  be  equal  to  the  degree  of 
the  differential  equation  expressing  its  rate. 

The  expressions  for  a  reaction  of  a  third  order  (between  three 
substances  A,  B,  and  C)  can  be  similarly  derived.  For  the  case  that 
the  three  substances  have  the  same  initial  concentration  C0  the  inte- 

grated expression  is  :  -   ---  —  2  k  t. 

(  n  r<\2  r<  2 

(C0  —  C)  C0 

Pro&.  7.  When  0.01  mol  of  methyl  acetate  is  dissolved  at  25°  in 
1  1.  of  0.01  normal  NaOH,  11.8%  of  the  ester  is  decomposed  per 
minute  at  the  start  according  to  the  reaction  :  CH3Ac  -f-  NaOH  — 
NaAc  +  CH,OH.  a.  How  long  will  it  take  for  one  half  of  the  ester 
in  this  mixture  to  be  saponified?  6.  How  long  will  it  take  when 
0.01  mol  is  dissolved  in  1  liter  of  0.02  normal  NaOH?  c.  Suggest  an 
analytical  method  by  which  the  rate  of  this  reaction  could  be  followed 
experimentally. 

65.  Expressions  of  the  Law  of  Concentration-Effect  in  Terms  of 
the  Fraction  of  the  Reacting  Substances  Transformed. 

Profc.  9.  From  the  integrated  expressions  of  the  first,  second,  and 
third  orders  already  obtained  derive  the  expressions: 


in  which  x  represents  the  fraction  of  the  reacting  substances  trans- 
formed, v  the  volume  of  the  mixture,  and  NO  the  equivalents  of  each 
of  the  reacting  substances  present  at  the  start. 

Pro&.  10.  Show  from  the  expressions  derived  in  Prob.  9  how  the 
specific  reaction-rate  is  related  to  the  time  required  for  the  transforma- 
tion of  any  definite  fraction  of  the  reacting  substances  in  the  case  of 
different  reactions  of  the  same  order,  or  of  the  same  reaction  at  different 
temperatures. 

Prob.  11.  At  25°  the  specific  rate  of  the  reaction  between  sodium 
hydroxide  and  methyl  acetate  is  1.8  times  as  great  as  that  of  the  re- 
action between  sodium  hydroxide  and  ethyl  acetate.  What  is  the  ratio 
of  the  times  required  for  decomposing  90%  of  the  two  esters  when 
equivalent  quantities  are  present  at  the  start? 


EFFECT  OF  CONCENTRATION  63 

Prob.  12.  What  is  the  ratio  of  the  times  required  for  decomposing: 
a.  90%  of  1  mol  of  cane-sugar  when  it  is  dissolved  in  1  1.  and  in  10  1. 
of  0.1  normal  HC1  solution?  6.  90%  of  1  mol  of  methyl  acetate  when 
it  is  dissolved  in  1  1.  of  normal  NaOH  solution  and  in  10  1.  of  0.1  normal 
XaOH  solution? 

66.  The  Mechanism  of  Reactions.  Influence  of  the  Number  of 
Molecules  Involved.  —  The  rate  of  certain  reactions,  such  as: 
2FeCl3  -f  SnCl2  =  2FeCl2  +  Sn014,  and 
2AgAc  +  NaCHO2  =  2Ag  +  C02  +  HAc  +  NaAc, 
in  which  two  molecules  of  one  of  the  substances  are  involved,  has  been 
found  to  be  proportional  to  the  square  (instead  of  to  the  first  power) 
of  the  concentration  of  that  substance.  In  other  words,  it  is  found 
that  these  reactions  are  of  the  third  order,  instead  of  the  second  order. 
The  view  that  the  number  of  reacting  molecules  determines  the  law 
of  the  rate  is  further  substantiated  by  the  fact  that  it  harmonizes 
the  law  of  reaction-rate  with  the  well-established  law  of  chemical 
equilibrium  (see  Prob.  26).  These  considerations  justify  the  conclu- 
sion that  the  provisional  statement  of  the  law  of  concentration-effect 
should  be  modified  so  as  to  state  that  the  rate  is  proportional  to  the 
concentration  of  each  of  the  reacting  substances  raised  to  a  power 
equal  to  the  number  of  its  molecules  which  interact  with  the  mole- 
cules of  the  other  substance  in  the  ultimate  molecular  process  on 
which  the  occurrence  of  the  reaction  depends.  Thus,  in  the  case  of  a 
reaction  aA  -f-  frB  -f-  cC  =  .  .  .,  whose  occurrence  requires  the  inter- 
action of  a  molecules  of  A,  b  molecules  of  B,  c  molecules  of  C,  the 
general  law  of  concentration-effect  is  expressed  by  the  equation: 

dc 

—  C)a  (COB  ~  C)6  (C0c  ~  G)C.  .  . 


Pro  6.  13.  Equal  volumes  of  0.2  normal  solutions  of  silver  acetate 
and  sodium  formate  were  mixed  at  100°  ;  and  after  definite  intervals 
of  time  samples  were  removed  and  the  undecomposed  silver  acetate 
was  titrated  with  potassium  thiocyanate.  Its  concentration  was  found 
to  be  0.067  normal  after  2  minutes,  0.047  normal  after  6  minutes,  and 
0.032  normal  after  14  minutes.  Show  from  these  data  that  this  re- 
action conforms  more  closely  to  the  expression  of  the  third  order  than 
to  that  of  the  second  order. 

It  is  found,  however,  that  many  reactions  which  apparently 
involve  three  or  more  molecules  conform  to  the  expression  of  the 
second  order.  This  is  probably  to  be  explained  by  the  assumption  that 


64  RATE  OF  CHEMICAL  REACTIONS 

the  reaction  expressed  by  the  usual  chemical  equation  takes  place  in 
stages,  and  that  the  stage  which  requires  appreciable  time  is  a  reaction 
between  two  molecules.  Thus  the  second-order  reaction  H2O2  +  2HI 
=  2H2O  -f-  I2  may  be  considered  to  take  place  in  the  two  stages, 
H2O2  +  HI  =  H20  +  HIO,  and  HIO  +  HI  =  H2O  +  I2;  the 
first  requiring  a  measurable  time,  and  the  second  taking  place  almost 
instantaneously  as  soon  as  any  HIO  is  formed  by  the  first  reaction. 
It  is  therefore  necessary  in  the  case  of  complex  reactions  to  know  their 
mechanism  (the  molecular  process  by  which  they  take  place),  in  order 
to  predict  the  law  of  their  rate;  and  conversely,  the  law  of  the  reaction- 
rate  throws  light  on  the  mechanism  of  the  reaction. 

Pro 6.  14.  a.  Suggest  an  explanation  of  the  fact  that  the  rate  of 
the  reaction  H+BrO8-  -f  6H+I-  =  H+Bi-  -j-  3I2  -f  3H2O  is  propor- 
tional to  the  first  power  of  the  concentration  of  the  BrO8-  and  of  that 
of  the  I-.  &.  Explain  also  the  fact  that  the  decomposition  of  arsine  gas, 
4AsH,  =  As4  -|-  6H2,  is  a  reaction  of  the  first  order. 

A  knowledge  of  the  mechanism  of  reactions  is  often  important  \n 
other  ways.  Thus,  the  initial  rates*  at  which  an  ester  is  decomposed 
by  different  bases  of  the  same  concentration  are  found  to  be  propor- 
tional to  the  degrees  of  ionization  of  the  bases,  showing  that  it  is  the 
hydroxide-ion  which  is  directly  involved  in  the  reaction;  and  IJiis 
knowledge  makes  it  possible  to  calculate  the  rate  of  decomposition 
of  an  ester  by  any  solution  whose  hydroxide-ion  concentration  is 
known,  or  to  make  the  converse  calculation. 

Pro&.  15.  At  25°  the  initial  rate  of  decomposition  of  ethyl  acetate 
by  0.01  normal  NaOH  is  9.0  times  as  great  as  that  by  0.1  normal  KGN. 
The  sodium  hydroxide  is  96%  ionized.  What  is  the  concentration  of 
hydroxide-ion  in  the  potassium  cyanide  solution? 

*The  initial  rates  have  to  be  considered  for  the  reason  that  the  neutral 
salt  which  is  produced  as  the  reaction  progresses  has  a  great  influence  on  the 
ionization  of  slightly  ionized  bases,  as  will  be  explained  in  the  next  chapter. 


EFFECT  OF  CATALYZERS  66 

THE  EFFECT  OF  CATALYZERS 

67.  Catalysis  and  Catalyzers. — A  reaction  is  often  greatly  accel- 
erated by  the  presence  of  a  substance  which  is  not  itself  consumed 
by  the  reaction.    This  phenomenon  is  called  catalysis,  and  the  sub- 
stance producing  it  is  called  the  catalyzer. 

Although  few  general  principles  relating  to  catalysis  have  been 
established,  its  great  practical  importance  makes  it  desirable  to  con- 
sider the  more  common  types  of  catalyzers  and  the  ways  in  which 
they  act.  This  is  done  in  Arts.  68-72. 

68.  Carriers. — Carriers  constitute  one  of  the  most  common  and 
best  understood  types  of  catalyzers.    The  mechanism  of  their  action 
is  as  follows:  The  catalyzer  produces  with  one  of  the  substances  an 
intermediate  compound  which  reacts  with  the  second  substance  in  such 
a  way  as  to  regenerate  the  catalyzer;  the  reaction  of  the  second  sub- 
stance with  the  intermediate  compound  taking  place  more   rapidly 
than  that  with  the  first  substance.    In  this  way  a  reaction  which  does 
not  take  place  directly  at  an  appreciable  rate  may  be  made  to  take 
place  in  stages  at  a  rapid  rate.    The  chamber  process  of  making  sul- 
phuric acid  and  the  technical  method  of  making  ether  are  familiar 
examples  of  this  type,  the  fundamental  reactions  being: 

a.  O2  +  2NO  =  2NO2,  and  SO2  +  NO2  +  H2O  =  H2SO4  +  NO. 

b.  C2H5OH  -f  H2SO4  =  C2H5HSO4  +  H2O,  and 
C2H5OH  -f  C2H5HS04  =  (C2H5)20  +  H2SO4. 

69.  Contact  Agents. — Reactions  between  gases  or  solutes  are  often 
greatly  accelerated  by  placing  the  reacting  mixture  in  contact  with 
a  suitable  solid  substance  which  offers  a  large  surface.    The  heavier 
metals  are  especially  likely  to  be  effective;  but  many  other  substances 
have   specific  effects   on   definite   reactions.     The  platinum   contact- 
process  of  making  sulphur  trioxide  from  sulphur  dioxide  and  oxygen 
and   the   Deacon  process   of  making   chlorine   by   passing   hydrogen 
chloride  and  oxygen  over  a  porous  mass  impregnated  with  copper 
chloride  are  examples  of  contact  catalysis.    Gas  reactions  are  often 
catalyzed  by  solid  surfaces  to  such  an  extent  that  the  chemical  change 
takes  place  appreciably  only  in  the  gas  that  is  in  immediate  contact 
with  the  walls  of  the  containing  vessel  or  with  solid  material  with 
which  it  may  be  charged. 

The  mechanism  of  contact  actions  is  little  understood.    In  most 


66  RATE  Of   CHEMICAL  REACTIONS 

cases,  the  contact  action  is  probably  due  to  an  adsorption  of  the  re- 
acting substances  (that  is,  to  a  concentration  of  them  on  the  surface 
of  the  solid)  and  to  the  fact  that  in  the  surface-layer  the  reaction - 
rate  is  greatly  increased.  Thus,  finely  divided  platinum  placed  in 
contact  with  illuminating  gas  and  air  adsorbs  a  large  quantity  of  the 
gases,  and  these  then  react  so  rapidly  as  to  cause  the  gas  to  take  fire. 

To  reactions  brought  about  by  a  contact  agent  the  law  of  concen- 
tration-effect is,  in  general,  not  applicable;  for  the  rate  of  such 
reactions  must  evidently  be  influenced  by  the  rates  at  which  the  re- 
acting substances  are  adsorbed  by  the  solid  surface. 

70.  Hydrogen-Ion  and  Hydroxide-Ion  as  Catalyzers. — In  aqueous 
solutions  many  reactions  are  accelerated  by  hydrogen-ion.  This  is 
probably  true  of  all  reactions  in  which  water  is  directly  involved, 
such  as  the  hydrolysis  of  cane-sugar  or  of  esters.  It  is  also  true  of 
certain  reactions  of  oxidation  and  reduction. 

The  rate  of  such  hydrolytic  reactions  in  very  dilute  solutions  is 
found  to  be  proportional  to  the  concentration  of  the  hydrogen-ion; 
for  example,  the  specific  reaction-rate  of  the  cane-sugar  hydrolysis 
at  48°  has  been  found  to  be  9.95  times  as  great  in  0.01  normal  HC1 
as  it  is  in  0.001  normal  HC1,  while  the  ratio  of  the  hydrogen-ion 
concentrations  in  the  two  solutions  is  9.8.  At  higher  hydrogen-ion 
concentrations  or  in  the  presence  of  neutral  salts  considerable  devia- 
tions from  proportionality  exist;  thus,  the  rate  of  the  cane-sugar 
hydrolysis  is  10.5  times  as  great  in  0.1  normal  HC1  as  in  0.01  normal 
HC1,  while  the  hydrogen-ion  concentration  is  only  9.5  times  as  great. 

This  principle  can  be  employed  (as  shown  by  Prob.  18)  for  de- 
termining the  hydrogen-ion  concentration  in  solutions;  for  no  other 
ion  (except  hydroxide-ion  in  certain  cases)  exerts  a  catalytic  effect 
on  hydrolytic  reactions. 

Reactions  in  which  water  takes  part  are  often  accelerated  also  by 
hydroxide-ion;  thus,  milk-sugar,  CJIaOu,  dissolved  in  water  becomes 
hydrated  (with  formation  of  C12H22OU.H2O)  at  a  rate  which  is  greatly 
increased  by  hydroxide-ion. 

Prob.  16.  Name  two  or  three  other  reactions,  besides  those  here 
referred  to,  which  are  accelerated  by  hydrogen-ion.  Refer,  if  necessary, 
to  chemical  text-books. 

Prob.  n.  Show  how  the  catalytic  effect  of  hydrogen-ion  on  hydro- 
lytic reactions  may  be  interpreted  as  a  carrier  action,  assuming  that 
the  hydrogen-ion  is  hydrated. 


EFFECT  OF  TEMPERATURE  67 

Prob.  18.  Diazoacetic  ester  decomposes  in  aqueous  solution  accord- 
ing to  the  equation  CHN2.CO2C2H5  4.  H2O  =  CH2OH.CO2C2H5  -f  N2, 
and  the  reaction  is  catalyzed  by  hydrogen-ion.  At  25°  in  a  solution  0.1 
normal  in  acetic  acid,  whose  ionization  is  1.34%,  37.5%  of  the  ester  is 
decomposed  in  10  minutes.  Assuming  that  it  takes  67  minutes  to  decom- 
pose the  same  percentage  of  the  ester  in  a  solution  0.1  formal  in  sodium 
hydrogen  tartrate,  what  is  the  hydrogen-ion  concentration  in  that 
solution? 

71.  Enzymes. — Certain  complex  organic  substances  called  enzymes, 
which  are  produced  by  animal  and  plant  organisms,  have  an  extraor- 
dinary power  of  catalyzing  certain  organic  reactions.    The  effect  is 
highly  specific,  a  particular  enzyme  being  required  for  a  particular 
reaction.    Thus  invertase,  an  enzyme  produced  by  yeast,  causes  the 
conversion   of   cane-sugar   into   glucose   and   fructose;    and    xymase, 
another  yeast  enzyme,  causes  the  conversion  of  glucose,  but  not  of 
the   analogous    compound   fructose,    into   ethyl    alcohol    and    carbon 
dioxide. 

72.  Water  as  a  Catalyzer.— The  presence  of  water  in   at  least 
minute  quantity  is  essential  to  the  occurrence  of  almost  all  reactions. 
This  is  shown  by  experiments  upon  some  of  the  most  energetic  chem- 
ical changes,  such  as  the  combination  of  sodium  with  chlorine,  the 
union  of  ammonia  and  hydrogen  chloride  gases,  the  combustion  of 
carbon  monoxide  with  oxygen,  and  the  union  of  lime  and  sulphur 
tri  oxide,  which  are  found  not  to  take  place  when  the  separate  sub- 
stances are  very  thoroughly  dried  before  they  are  brought  together. 


THE   EFFECT  OF  TEMPERATURE 

73.  The  Law  of  Temperature-Effect.— Equal  small  increments  of 
temperature  cause  an  approximately  equal  multiplication  of  the 
specific  rate  of  any  definite  reaction.  Thus,  if  the  specific  rate  of  a 
reaction  is  increased  2.5-fold  by  raising  the  temperature  from  0  to  10°, 
it  will  again  be  increased  approximately  2.5-fold  by  raising  the  tem- 
perature from  10  to  20°. 

The  deviations  from  this  law  are  greater,  the  greater  the  interval 
of  temperature  to  which  it  is  applied.  The  general  magnitude  of 
them  is  illustrated  by  the  ratios  of  the  specific  reaction-rates  at  10° 
intervals  for  the  third-order  reaction  between  ferrous  chloride,  potas- 
sium chlorate,  and  hydrochloric  acid,  which  have  been  found  to  be 


68  RATE  OF  CHEMICAL  REACTIONS 

as  follows:  2.8  between  0  and  10°,  2.7  between  10  and  20°,  2.4  between 
20  and  30°,  2.5  between  30  and  40°,  and  2.2  between  40  and  50°. 

Prob.  19.  In  a  solution  0.01  molal  in  sodium  hydroxide  and  0.01 
molal  in  ethyl  acetate,  39%  of  the  ethyl  acetate  is  decomposed  in 
10  minutes  at  25°,  and  55%  at  35°.  How  long  would  it  take  to  decom- 
pose 55%  at  15°? 

Prob.  20.  A  more  general  expression  of  the  law  of  temperature- 
effect  Is  that  given  by  the  equation  dlogfc  —  (A/T*)dT,  in  which  A  is 
a  quantity  which  has  nearly  the  same  value  at  temperatures  not  far 
apart.  Integrate  this  equation,  and  show  that  the  result  for  a  small 
range  of  temperature  is  in  accordance  with  the  above  stated  law  of 
temperature-effect. 

Prob.  21.  Calculate  by  this  more  general  expression  from  the  data 
of  Prob.  19  how  long  it  would  take  to  decompose  50%  of  the  ethyl 
acetate  at  20°. 

With  respect  to  the  effect  of  temperature,  it  is  further  to  be  noted 
that  in  the  case  of  different  reactions  equal  small  increments  of  tem- 
perature cause  not  far  from  the  same  multiplication  of  their  specific 
rates.  Thus,  a  10°  rise  of  temperature  multiplies  the  specific  rates 
of  the  reactions  between  the  following  substances  by  the  following 
factors : 

NaOH  -f-  C,H5C2H302   (at  27°)  :  1.9 

FeSO4  -f  KC103  -f  H2S04  (at  21°)  :  2.4 
NaOH  -f-  CH2ClCO2Na  (at  100°)  :  2.5 
C^H^O,,  -f  H20  (+  HC1)  (at  40°)  :  3.6 

Prob.  22.  At  100°  a  certain  reaction  takes  place  to  an  extent  of  25% 
In  one  hour.  Estimate  roughly  how  long  it  would  take  for  the  reaction 
to  proceed  to  the  same  extent  at  20°. 


THE    EFFECT    OF    SURFACE    IN    THE    CASE    OF    SOLID    SUBSTANCES 

74.  Solid  Substances  Involved  in  Chemical  Reactions.— When  a 
solid  substance  is  reacting  with  a  solute,  the  quantity  of  it  dissolved 
per  unit  of  time  is  proportional  to  the  surface  of  the  solid,  as  well  as 
to  the  concentration  of  the  solute  that  reacts  with  it.  Thus,  when 
a  dilute  solution  of  acetic  acid  in  contact  with  a  compact  mass  of 
magnesium  oxide  is  uniformly  stirred,  the  quantity  of  the  oxide 
dissolved  is  proportional  to  the  surface  of  the  mass  and  to  the  con- 
centration of  the  acetic  acid. 


SIMULTANEOUS  REACTIONS  69 

In  reactions  with  solid  substances,  it  is  to  be  borne  in  mind  that, 
owing  to  corrosion,  the  effective  surface  is  constantly  changing,  and 
that  the  concentration  of  the  solution  in  contact  with  the  solid  is 
the  same  as  that  of  the  whole  solution  only  when  there  is  adequate 
stirring. 

75.  Solid  Substances  Dissolving  in  Their  Own  Solutionr. — When 
a  solid  substance  is  dissolving  at  a  definite  temperature  in  its  own 
(partially  saturated)  solution,  the  quantity  dissolved  per  unit  of  time 
is  proportional  to  the  surface  of  the  solid  and  to  the  difference  between 
the  concentration  of  the  saturated  solution  and  that  of  the  solution 
in  contact  with  the  solid. 

J'rob.  23.  A  quantity  of  a  solid  substance  having  a  surface-area  a 
and  a  solubility  «  is  stirred  uniformly  at  a  definite  temperature  with 
a  definite  volume  <;  of  water,  a.  Formulate  a  differential  expression  for 
the  rate  at  which  the  concentration  c  of  the  solution  will  increase. 
I).  Integrate  this  expression,  assuming  the  surface  to  remain  constant 
(which  it  does  approximately  when  the  quantity  of  the  solid  substance 
is  very  large  in  relation  to  the  quantity  of  it  that  dissolves). 

J'rob.  g.'i.  a.  What  are  the  relative  rates  at  which  *a  substance  dis- 
solves in  its  own  solution  when  the  solution  is  50%  and  95%  saturated, 
assuming  the  surface  of  the  solid  and  the  volume  of  the  solution  to  be 
the  same  in  the  two  cases?  It.  Tf  it  takes  2  minutes  for  the  solution 
to  become  50%  saturated,  how  long  will  it  take  for  the  degree  of  satura- 
tion to  increase  from  95  to  98%  ? 

SIMULTANEOUS  REACTIONS 

76.  Law  of  Independent  Reaction-Rates. — When  two  reactions  are 
taking  place  simultaneously,  the  rate  of  each  is  determined  by  its 
own  specific  reaction-rate  and  by  the  concentrations  of  the  substances 
involved  in  it,  just  as  if  the  other  reaction  were  not  taking  place. 

Pi-ob.  25.  The  reaction  between  methyl  oxalate  and  sodium  hydrox- 
ide takes  place  in  the  two  stages : 

(CH8)2C204  +  NaOH  =  CH8NaC2O4  +  CH.OH 
CH8NaC204  +  NaOH  =  Na2C2O4  +  CH.OH. 

Formulate  the  differential  equation  for  the  rate  (expressed  in  molal 
concentration)  of  each  of  these  reactions  for  the  case  that  the  methyl 
oxalate  and  sodium  hydroxide  are  present  at  the  start  at  the  molal 
concentrations  CA  and  CB,  respectively,  and  that  at  the  time  t  the 
sodium  methyl  oxalate  and  the  sodium  oxalate  produced  have  attained 
the  molal  concentrations  c^  and  c2,  respectively. 


70  RATE  OF  CHEMICAL  REACTIONS 

77.  Reactions  Taking  Place  in  Opposite  Directions. — The  law  of 
independent  reaction-rates  finds  an  especially  important  application 
in  the  case  of  reactions  which  do  not  go  to  completion,  but  take 
place  in  one  direction  when  the  substances  on  the  one  side  of  the 
chemical  equation  are  brought  together,  and  in  the  other  direction 
when  those  on  the  other  side  are  brought  together.  The  resultant  rate 
j)f  such  a  reaction  must  evidently  be  the  difference  of  the  rates  of  the 
two  opposing  reactions. 

Prob.  26.  The  gaseous  reaction  H2  -f-  I2  =  2HI  is  an  incomplete 
reaction  of  the  kind  just  described,  which  at  400-500°  takes  place  at  a 
measurable  rate.  For  the  case  that  hydrogen,  iodine,  and  hydrogen 
iodide  are  present  at  the  start  at  the  molal  concentrations  COH,  cOI) 
and  cOHi»  formulate  the  differential  expression  for  the  rate  at  which 
the  molal  concentration  of  the  hydrogen  iodide  is  increasing  at  the 
time  *,  when  c  mols  per  liter  of  hydrogen  iodide  have  been  produced. 


CHAPTEK    VI 

THE  EQUILIBKIUM  AT  CONSTANT  TEMPEKATUEE  OF 
CHEMICAL  KEACTIONS  INVOLVING  GASEOUS  OK  DIS- 
SOLVED SUBSTANCES  AT  SMALL  CONCENTEATIONS 


THE  LAW  OP  MASS-ACTION 

78.  The  Equilibrium  of  Chemical  Reactions. — When  two  or  more 
chemical  substances  capable  of  reacting  with  one  another  are  brought 
together,  it  is  always  true  after  a  sufficiently  long  time  (which  may 
vary  from  a  fraction  of  a  second  to  thousands  of  years)  that  the 
chemical  reaction  which  has  been  taking  place  between  them  prac- 
tically ceases — in  other  words,  that  a  condition  is  reached  where 
no  further  change  takes  place.  The  reaction  is  then  said  to  be  in 
equilibrium. 

Many  reactions  (for  example,  the  reaction  NH4OH  -\-  HBO2  = 
NH4BO2  -f-  H2O  in  aqueous  solution)  take  place  so  incompletely 
.that  at  equilibrium  the  substances  on  both  sides  of  the  equation  are 
present  in  measurable  proportions.  But  with  many  other  reactions 
the  equilibrium-conditions  are  such  that  the  change  seems  to  take 
place  completely  when  the  substances  on  one  side  of  the  chemical 
equation  expressing  it  are  brought  together,  and  not  to  take  place 
at  all  when  the  substances  on  the  other  side  are  brought  together; 
thus,  this  is  true  of  the  reaction  NaOH  +  HC1  =  NaCl  +  H2O 
in  aqueous  solution.  In  all  such  cases,  however,  the  gaseous  or  dis- 
solved substances  on  both  sides  of  the  equation  are  really  present  at 
some  concentration,  though  this  may  be  so  small  in  the  case  of  some 
of  the  substances  that  it  cannot  be  directly  measured. 

Keactions  (like  that  in  a  mixture  of  hydrogen  and  oxygen  at  25°) 
which  are  taking  place  so  slowly  that  no  appreciable  change  can  be 
detected  within  a  reasonable  time  are  not  to  be  confounded  with 
those  which  are  in  a  state  of  equilibrium.  Whether  equilibrium- 
conditions  have  been  attained  can  be.  determined  by  causing  the 
reaction  to  take  place  in  the  two  opposite  directions  and  comparing 

71 


72  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

the  concentrations  of  the  substances  in  the  two  cases  after  equi- 
librium seems  to  have  been  reached. 

The  equilibrium-conditions  of  chemical  reactions  vary  with  the 
temperature.  In  this  chapter  and  the  following  one  the  principles 
will  be  discussed  which  determine  equilibrium  when  the  temperature 
has  any  constant  value. 

79.  The  Mass-Action  Law. — The  effect  of  concentration  in  deter- 
mining chemical  equilibrium  is  expressed  by  one  of  the  most  funda- 
mental laws  of  chemistry.  This  law,  which  is  commonly  known  as 
the  mass-action  law,  states  that,  whatever  be  the  initial  concentra- 
tions of  the  gaseous  or  dissolved  substances  A,  B,  .  .  E,  F,  .  .  involved 
in  any  definite  chemical  reaction,  such  as  may  be  represented  in 
general  by  the  equation,  flA  -f-  bB . .  =  eE  -j-  /F . . ,  the  reaction 
always  takes  place  in  such  a  direction  and  to  such  an  extent  that, 
when  equilibrium  is  reached  at  any  definite  temperature,  the  condi- 
tions are  satisfied  which  are  expressed  by  the  equation : 

-  =  K,  a  constant. 
cAac*b  .  . 

In  this  expression  CE,  CF,  .  .  CA,  CB,  .  .  denote  the  concentrations  of 
the  substances  E,  F, . .  A,  B, . .  in  the  equilibrium  mixture,  and 
e,  f, . .  a,  fc, . .  denote  the  number  of  mols  of  them  that  are  involved 
in  the  reaction  expressed  by  the  chemical  equation. 

The  quantity  K,  which  is  a  constant  characteristic  of  the  reaction, 
is  called  its  equilibrium-constant.  Its  value  is,  of  course,  constant 
with  respect  to  variations  of  the  initial  and  equilibrium  concentra- 
tions, but  it  varies  with  the  temperature.  In  evaluating  it,  it  is 
customary  to  express  all  concentrations  in  mols  per  liter,  and  to  place 
in  the  numerator  the  concentrations  of  the  substances  occurring  on 
the  right-hand  side  of  the  chemical  equation  written  in  some  speci- 
fied way.  In  which  direction  the  chemical  equation  is  written  is,  of 
course,  arbitrary  in  the  case  of  reactions  at  equilibrium.  With  many 
types  of  reactions,  however,  it  is  more  natural  to  write  the  equation 
in  one  of  the  two  directions;  and  in  such  cases  the  usage  in  evalu- 
ating the  equilibrium-constant  has  become  fairly  definite.  This  usage 
will  be  illustrated  by  the  examples  given  in  the  following  articles. 

The  mass-action  law  has  been  verified  by  direct  experiments  upon 
a  large  number  of  different  reactions.  It  can  be  derivpd,  as  will  bf> 


BETWEEN  GASEOUS  SUBSTANCES  73 

shown  in  a  later  chapter,  with  the  aid  of  the  laws  of  thermodynamics, 
from  the  perfect-gas  equation  (p  =  c  R  T),  or  from  the  osmotic- 
pressure  equation  for  perfect  solutes  (P  =  cRT).  It  can  be  also 
derived  from  the  laws  of  the  rate  of  reactions,  as  is  illustrated  by 
Prob.  1  below. 

The  mass-action  law  is  exact  only  in  the  case  of  reactions  between 
perfect  gases  or  perfect  solutes;  but  it  holds  true  approximately 
when  applied  to  gases  at  moderate  pressures  or  to  unionized  or  slightly 
ionized  solutes  at  moderate  concentrations.  The  deviations  from  it 
in  such  cases  may  be  expected  to  be  of  the  same  order  of  magnitude 
as  those  from  the  pressure-concentration  law  of  perfect  gases  or 
perfect  solutes.  As  will  be  seen  later,  large  deviations  are  met  with 
in  the  case  of  largely  ionized  substances. 

J'rob.  1.  a.  Formulate  the  mass-action  expression  for  the  equilib- 
rium of  the  reaction  2HI  —  H2  -f  I2,  and  derive  it  from  the  differential 
equation  obtained  in  Prob.  26  of  the  previous  chapter,  explaining  the 
substitutions  that  are  made  in  the  derivation.  6.  Show  what  relation 
exists  between  the  equilibrium-constant  and  the  specific  reaction-rates. 

Note. — Although  the  law  of  chemical  equilibrium  can  be  derived,  as 
here  illustrated,  from  the  law  of  reaction-rate,  an  important  difference 
between  the  two  laws  is  to  be  noted.  Namely,  it  can  be  shown  that, 
although  the  expression  for  the  rate  of  a  reaction  depends  on  its 
mechanism,  the  same  expression  is  obtained  for  its  equilibrium,  what- 
ever be  the  process  by  which  it  is  considered  to  be  attained.  In  using 
the  equilibrium  expression  it  is  only  necessary  to  know  the  concentra- 
tions of  the  substances  actually  occurring  in  that  expression. 

APPLICATIONS   OF  THE   MASS-ACTION   LAW  TO   GASEOUS   SUBSTANCES 

80.   Expression  of  the  Law  in  Terms  of  Partial  Pressures. — In 

applications  of  the  mass-action  law  to  gases,  it  is  usual  to  substitute 
for  the  concentrations  of  the  substances  in  the  equilibrium-mixture 
their  partial  pressures.  This  is  admissible  since  at  any  definite 
temperature  the  two  quantities  are  proportional  to  one  another,  in 
virtue  of  the  relation  p  =  c  R  T. 

The  general  mass-action  expression  in  terms  of  partial  pressures 
evidently  is 


where  Kp  is  a  constant,  called  the  equilibrium-constant  in  terms  of 


74  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

pressure,  which  has  in  general  a  different  numerical  value  from  the 
constant  K  occurring  in  the  corresponding  concentration-expression. 
In  evaluating  it  the  partial  pressures  are  commonly  expressed  in 


Prob.  2.  Derive  the  general  mass-action  expression  in  terms  of 
pressure  from  that  in  terms  of  concentration,  and  show  the  relation 
between  the  two  equilibrium-constants. 

81.  Gaseous  Dissociation. — A  chemical  change  which  consists  in 
the  splitting  of  a  substance  with  complex  molecules  into  one  or  more 
substances  with  simpler  molecules  is  called  dissociation.  Thus,  the 
reactions  I2  =  21,  NH4C1  =  NH3  +  HC1,  and  2CO2  =  SCO  +  O2, 
are  examples  of  dissociation.  The  fractional  extent  to  which  the 
dissociating  substance  has  been  decomposed  is  called  its  degree  of 
dissociation,  or  simply  its  dissociation  (7).  The  equilibrium-constant 
of  such  a  reaction  is  commonly  called  the  dissociation-constant  of 
the  dissociating  substance. 

It  is  characteristic  of  such  reactions  that  the  number  of  mole- 
cules increases  when  the  dissociation  increases.  Since  the  pressure- 
volume  product  of  gases  increases  correspondingly,  the  degree  of 
dissociation  can  always  be  determined  by  measuring  the  volume 
or  density  of  the  gas  at  a  known  temperature  and  pressure,  and 
comparing  it  with  that  calculated  for  the  undissociated  or  completely 
dissociated  substance,  as  is  illustrated  by  Prob.  14,  Art.  17,  and  by 
Prob.  4  below. 

Another  common  method  of  determining  the  composition  of  the 
equilibrium-mixture  is  to  cool  it  suddenly  to  a  lower  temperature 
at  which  the  reaction-rate  is  so  small  that  the  original  equilibrium  is 
not  displaced,  and  then  to  analyze  the  mixture.  This  method  pre- 
supposes that  there  is  no  change  in  the  composition  during  the  short 
period  of  cooling.  This  condition  may  be  practically  realized  in  cases 
where  the  rate  at  which  the  equilibrium  is  established  is  compara- 
tively slow  even  at  the  higher  temperature,  or  in  cases  where  the 
reaction  takes  place  only  in  contact  with  a  catalyzer.  lu  the  latter 
case  the  equilibrium-mixture  can  be  separated  from  the  catalyzer, 
and  subsequently  cooled  without  the  danger  of  any  change  taking 
place. 

Prob.  8.  a.  Formulate  the  mass-action  expression  for  the  dissocia- 
tion of  sulphur  trioxide  into  sulphur  dioxide  and  oxygen.  6.  At  630° 


BETWEEN  GASEOUS  SUBSTANCES  75 

and  1  atni.  the  sulphur  trioxide  is  just  one  third  dissociated.    Calculate 
the  dissociation-constant  of  sulphur  trioxide. 

Prob.  4.  Show  how  the  dissociation  (7)  of  sulphur  trioxide  could 
be  calculated  from  measurements  of  the  density  (d)  of  the  equilibrium 
mixture  at  the  temperature  (T)  and  pressure  (p)  in  question. 

Pro 6.  5.  A  mixture  consisting  of  1  mol  of  SO2  and  1  mol  of  O2  is 
passed  at  630°  and  1  atm.  through  a  tube  containing  finely  divided 
platinum  so  slowly  that  equilibrium  is  attained,  and  the  issuing  gas  is 
cooled  and  analyzed  by  absorbing  the  sulphur  dioxide  and  trioxide 
by  potassium  hydroxide  and  measuring  the  residual  oxygen  gas.  At  0° 
and  1  atm.  the  volume  of  this  residual  gas  was  found  to  be  13,780  cc., 
corresponding  to  0.615  mol.  a.  Calculate  the  dissociation-constant  of 
sulphur  trioxide.  b.  Calculate  the  ratio  of  the  mols  of  sulphur  trioxide 
to  the  mols  of  sulphur  dioxide  in  an  equilibrium  mixture  at  630°  in 
which  the  partial  pressure  of  oxygen  is  0.25  atm. 

/Yob.  6.  At  a  certain  temperature  a  definite  quantity  of  phosphorus 
pentachloride  gas  has  at  1  atm.  a  volume  of  1  liter,  and  under  these 
conditions  it  is  about  50%  dissociated  into  PC18  and  C12.  Show  by 
reference  to  the  mass-action  expression  whether  the  dissociation  will 
be  increased  or  decreased:  a,  when  the  pressure  on  the  gas  is  reduced 
till  the  volume  becomes  21.;  &,  when  nitrogen  is  mixed  with  the  gas 
till  the  volume  becomes  2  1.,  the  pressure  being  still  1  atm.;  c,  when 
nitrogen  is  mixed  with  the  gas  till  the  pressure  becomes  2  atm.,  the 
volume  being  still  1  1. ;  d,  when  chlorine  is  mixed  with  the  gas  till 
the  pressure  becomes  2  atm.,  the  volume  being  still  1  1. — In  answering 
these  questions  consider  whether  the  first  effect  of  the  change  in  con- 
ditions (assuming  that  no  reaction  takes  place)  is  to  increase  or  de- 
crease the  value  of  the  ratio  Pc\2  Ppc\s/Ppc\5>  and  in  which  direction 
the  reaction  must  take^  place  in  order  to  restore  the  equilibrium- value 
of  this  ratio. 

Prob.  7.    a.  Derive  for  the  dissociation  of  water- vapor  into  hydro- 
gen and  oxygen  a  mass-action   expression   which   will   show  how   the 
dissociation   (7)   varies  with  the  total  pressure   (p).     b.    At  2,000°  the 
dissociation  is  2.0%  when  the  total  pressure  is  1  atm.    How  much  is 
the  dissociation  when  the  total  pressure  is  0.2  atm.   (as  it  is  approxi 
mately  in  the  gaseous  mixture  produced  by  burning  hydrogen  with  the 
minimum  amount  of  air)?    Solve  the  equation  approximately,  neglect 
ing  the  (small)  value  of  the  dissociation  where  this  is  justifiable. 

82.  Metathetical  Gas  Reactions. — Examples  of  such  reaction* 
whose  equilibrium  has  been  investigated  are:  2HI  =  H,  +  I2*; 
CO2  +  H2  =  CO  +  H2O;  and  4HC!  +  O2  =  2C12  +  2H2O. 

*This  reaction  is  often  called  a  dissociation,  since  one  substance  is  con- 
verted into  two  others.  From  a  molecular  standpoint,  however,  it  consists  in 
an  interchange  of  atoms  between  molecules,  rather  than  in  a  splitting  of  a 
molocule  into  simpler  ones ;  and  from  a  mass-action  standpoint  it  differs  from 
true  dissociation  in  that  there  is  no  increase  in  the  number  of  molecules  when 
the  reaction  takes  place. 


76  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

An  important  principle  in  regard  to  them,  illustrated  by  Prob.  9 
below,  is  that  the  equilibrium-constant  of  any  metathetical  gas  re- 
action can  be  calculated  from  the  dissociation-constants  of  the  com- 
pounds involved  in  it. 

Another  important  principle  relates  to  the  effect  of  pressure.  It 
states  that  increase  of  pressure  causes  the  equilibrium  of  any  gaseous 
reaction  to  be  displaced  in  that  direction  in  which  the  number  of 
molecules,  and  therefore  the  volume  of  the  gas,  decreases.  This  prin- 
ciple has  already  been  illustrated  by  the  fact  that  dissociation  is 
decreased  by  increase  of  pressure.  It  is  demonstrated  by  Prob.  10. 

Prob.  8.  The  equilibrium-constant  of  the  reaction  CO2  -f  H2  = 
CO  -f  H2O  at  1,120°  is  2.0.  What  is  the  ratio  of  the  number  of  rnols 
of  CO2  and  of  H2O  that  will  be  formed  when  a  "water-gas"  consist- 
ing of  1  mol  CO  and  1  mol  H2  is  burnt  at  1,120°  with  $  mol  O2? 
(Assume  that  equilibrium  is  attained  and  that  the  quantity  of  the 
oxygen  which  remains  uncombined  is  negligible.) 

Prob.  9.  The  equilibrium  -constant  of  the  reaction  2CO2  =  2CO  -j-  O2 
at  1,120°  is  1.4  x  10~12.  a.  Calculate  the  partial  pressure  of  the  oxygen 
in  the  equilibrium  mixture  of  Prob.  8.  b.  Calculate  the  dissociation- 
constant  of  water-vapor  at  1,120°.  c.  Show  what  relation  exists  be- 
tween the  dissociation-constants  Kw  and  KC02  of  water-vapor  and 
of  carbon  dioxide  and  the  equilibrium-constant  K  of  the  reaction 
C02  -f  H2  =  CO  +  H2O. 

Prof).  10.  Prove  that  the  equilibrium  of  any  chemical  reaction, 
oA  -(-  6B  =  eE  -f-  /F,  must  be  displaced  in  the  direction  in  which  the 
number  of  molecules  decreases  when  the  total  pressure  p  of  the  equi- 
librium mixture  (in  which  the  substances  are  present  at  mol-fractions 
XA,  XB,  XE,  and  #F)  is  increased. 

APPLICATIONS  OP  THE  MASS-ACTION  LAW  TO  DISSOLVED  SUBSTANCES 

83.  lonization  01  Slightly  Ionized  Univalent  Acids  and  Bases. — 
The  mass-action  law  has  been  found  to  be  applicable  to  the  ionization 
(dissociation  into  ions)  of  the  slightly  ionized  monobasic  acids  and 
monacidic  bases.  This  is  illustrated  by  the  following  values  of  the 
percentage  ionization  at  18°  derived  from  conductance  measurements 
(as  described  in  Art.  55)  and  of  the  ionization-constant  K  of  ammo- 
nium hydroxide  (that  is,  the  equilibrium-constant  of  the  reaction 
NH4OH  ==  NH4+  +  OH") : 

Cone.  0.300  0.100  0.010  0.001 

1007  0.74  1.30  4.05  12.3 

106Z  16.6  17.1  17.1  17.1 


BETWEEN  DISSOLVED  SUBSTANCES  77 

The  ionization-constant  varies  greatly  with  the  composition  and 
structure  of  the  acid  or  base,  as  will  be  seen  from  the  following  values 
of  it  for  certain  acids  at  25°. 
Acid         1Q*K  Acid  10'K  Acid  Values  of  W«K 


HCN        0.0007 
HBO,       0.0007 
HC1O       0.04 
HNO,  500. 

HCO2H           210. 
CH3C02H         18. 
C2H5CO2H        13. 
n-C8H7CO2H     15. 

C6H6C02H 

C6H4OHC02H 
G6H4C1C02H 
C6H4N02C02H 

GO.       60.          60. 
Ortho     Mcta    Para 
1020.       87.       29. 
1320.     155.       93. 
6160.     345.     396. 

Prob.  11.  At  25°  acetic  acid  in  0.1  normal  solution  is  1.34%  ion- 
ized, a.  Calculate  the  ionization-constant  of  acetic  acid.  &.  Calculate 
its  ionization  in  0.01  normal  solution. 

Prob.  12. — Effect  of  the  Presence  of  a  Substance  with  a  Common 
Ion. — a.  Show  that  the  hydrogen-ion  concentration  in  an  acetic  acid 
solution  is  decreased  by  the  addition  of  sodium  acetate  approximately 
in  the  proportion  in  which  the  concentration  of  the  acetate-ion  is  in- 
creased, b.  State  the  corresponding  principle  that  would  apply  to  the 
ionization  of  ammonium  hydroxide  in  the  presence  of  an  ammonium 
salt.  c.  Calculate  the  ionization  of  acetic  acid  in  a  solution  0.1  normal 
both  in  acetic  acid  and  in  sodium  acetate,  the  sodium  acetate  in  the 
mixture  being  85%  ionized. — In  this  and  other  mass-action  problems 
make  any  simplifications  that  will  not  produce  in  the  result  an  error 
greater  than  1%. 

84.  Ionization  of  Dibasic  Acids  and  Their  Acid  Salts.— Polybasic 
acids  ionize  in  stages;  thus,  a  bibasic  acid  H2A  ionizes  according  to 
the  equations  H^  =  H+  +  HA~  and  HA"  =  H+  +  A=.  The  equi- 
librium-constants Kt  and  K2  of  these  reactions  are  called  the  ioniza- 
tion-constants  for  the  first  hydrogen  and  for  the  second  hydrogen, 
respectively.  The  values  of  K9  are  commonly  much  smaller  than 
those  of  Rr  The  values  at  25°  of  the  two  constants  for  some  impor- 
tant acids  are  as  follows : 

Acid  K^                                   Kz 

H2SO,  2  x  10-'  5  X  10-6 

H8P04  1  X  10-a  2  x  10-T 

H2C4H4O8(tartaric)  9.7  x  10-*  4  x  10-" 

H2CO,  3  x  10-'  3  x  10-" 

H2S  9  X  10-"  1  X  10-" 

From  the  two  ionization-constants  of  a  dibasic  acid  and  its  formal 
concentration  c  the  concentrations  of  the  various  substances  H,A, 
HA",  A=,  and  H*  present  in  its  solution  can  be  calculated.  In  solving 
any  such  mass-action  problem,  the  best  plan  is  to  formulate  firs* 


78  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

the  equilibrium  equations  that  must  be  satisfied — thus,  in  this  case 
the  equations  (IT)  (HA~)  ==  ^(H2A)  and  (IT)  (A~)  =  JC(HA-).* 
The  next  step  is  to  formulate  the  so-called  condition  equations,  which 
sum  up  the  concentrations  of  the  separate  forms  in  which  an  element 
or  other  constituent  exists  in  the  solution.  Thus,  in  this  case,  for  the 
total  molal  concentration  2  (A)  of  the  constituent  A,  and  for  that 
2  (H)  of  the  hydrogen,  we  have : 

(H2A)  +  (HA-)  +  (A=)  =  2  (A)  =  c, 
and  (H+)  -f  (HA-)  -f  2(H2A)  =  2(H)  =  2c. 

(It  will  be  noted  that  one  of  these  equations  might  be  replaced  by  a 
simpler  condition-equation  of  another  kind;  namely,  by  the  equation, 
(H*)  =  (HA~)  -f-  2(A=),  which  expresses  the  fact  that  positive  and 
negative  ions  must  be  present  in  equivalent  quantities.)  We  now 
have  four  independent  equations  containing  four  unknown  concen- 
trations. An  exact  algebraic  solution  of  these  equations  is  therefore 
possible.  But,  as  such  a  solution  is  often  very  complicated,  it  is 
advisable  to  try  first  to  simplify  the  condition-equations,  which  can 
often  be  done  by  neglecting  in  them  the  condensation  of  some  sub- 
stance which  is  small  in  comparison  with  the  concentrations  of  the 
other  substances.  Thus,  in  this  case  the  concentration  (A=)  is  small 
compared  with  the  concentration  (HA'),  for  the  reasons  that  K2  is 
small  in  comparison  with  K^  and  that  the  hydrogen-ion  produced 
by  the  ionization  of  the  H,A  into  H*  and  HA~  further  decreases  by 
the  common-ion  effect  the  ionization  of  the  HA~.  It  is  always  well 
to  test  the  correctness  of  such  simplifying  assumptions  by  subsequent 
calculation  of  the  quantity  neglected. 

Prob.  18.  From  the  ionization-constants  given  in  the  table  above, 
calculate  by  the  method  just  described  the  molal  concentration  of  each 
of  the  substances  present  in  a  0.01  formal  solution  of  tartaric  acid 
at  25°. 

Prob.  14. — Determination  of  the  lonization-Constant  for  the  Second 
Hydrogen  by  Reaction-Rate  Measurements.— By  measuring  the  conduct- 
ance of  tartaric  acid  in  0.06-0.01  formal  solution,  where  the  ioiiiza- 
tion  of  the  second  hydrogen  is  negligible,  the  ionization-constaut  for 
the  first  hydrogen  of  the  acid  has  been  found  to  be  0.00097  at  25°.  The 
hydrogen-ion  concentration  in  a  0.1  formal  solution  of  sodium  hydro 
gen  tartrate,  NaHA,  was  found  in  Prob.  18,  Art.  70,  to  be  0.0002  molal. 
From  analogy  with  other  salts  of  the  same  type  (see  Art.  58)  the 

*As  is  done  here,  the  molal  concentrations  of  substances  are  often  repre- 
sented by  writing  their  formulas  within  parentheses. 


BETWEEN  DISSOLVED  SUBSTANCES  79 

concentration  (NaHA)  of  unionized  sodium  hydrogen  tartrate  In  this 
solution  may  be  estimated  to  be  0.016  molal.  In  regard  to  the  concen- 
trations (Na2A)  and  '(NaA~)  nothing  definite  is  known,  but  they  may 
be  assumed  in  this  problem  to  be  negligibly  small.  From  these  data 
calculate  the  concentrations  (A=),  (HA-),  and  (H2A),  and  the  ioiriza- 
tion-constant  for  the  second  hydrogen  of  tartaric  acid.  (In  this  case 
no  simplification  of  the  condition-equations  is  admissible,  other  than 
the  neglecting  of  the  concentrations  (Na2A)  and  (NaA~)  just  men- 
tioned.) Tabulate  the  molal  concentrations  of  all  the  substances 
present  in  the  solution  of  the  sodium  hydrogen  tartrate. 

Prob.  15. — Determination  of  the  lonization-Constant  for  the  Second 
Hydrogen  by  Distribution  Experiments. — A  0.1  formal  solution  of 
sodium  hydrogen  succinate  NaHA  in  water  is  found  by  experiment 
to  be  in  equilibrium  at  25°  with  a  0.00187  formal  solution  of  succinic 
acid  in  ether.  The  distribution-ratio  of  succinic  acid  between  water 
and  ether  at  25°  is  7.5.  a.  Find  the  H2A  concentration  in  the  aqueous 
solution  of  the  acid  salt.  b.  Calculate  the  ionization-constant  for  the 
second  hydrogen.  That  for  the  first  hydrogen  is  6.6  X  10~B-  Make 
the  same  assumptions  as  in  Prob.  14;  neglect  also  the  value  of  (H+) 
in  the  condition-equation,  and  show  afterwards  by  calculating  it  that 
it  was  justifiable  to  neglect  it. 

85.  The  lonization-  of  Water. 

Prob.  16.  Show  that  the  product  (H+)  x  (OH-)  has  substantially 
the  same  value  in  any  dilute  aqueous  solution  (up  to  0.1-0.2  normal) 
as  in  pure  water.  (This  constant  value  of  the  (H+)  x  (OH~)  product 
is  called  the  ionization-constant  #w  of  water.) 

Note. — The  application,  involved  in  this  problem,  of  the  mass- 
action  law  to  a  constituent  so  concentrated  as  the  solvent  would  seem 
to  be  inadmissible,  and  is  so  in  a  strict  sense.  It  can  be  shown,  how- 
ever, with  the  aid  of  thermodynamics,  that  the  activity  of  the  solvent, 
that  is,  its  mass-action  effect  in  influencing  equilibrium,  is  exactly 
proportional  in  the  case  of  different  solutions  to  the  vapor-pressure 
of  the  solvent  above  those  solutions.  And  it  follows  from  Raoult's 
law  of  vapor-pressure  that  the  vapor-pressure  of  even  a  0.5  molal 
aqueous  solution  is  only  about  \%  less  than  that  of  pure  water. 

Prob.  17.  At  25°  the  concentration  of  hydroxide-ion  in  pure  water 
is  one  tenmillionth  molal.  What  is  its  concentration  in  0.1  formal 
HC1  solution,  in  which  the  acid  is  92%  ionized? 

86.  lonization  of  Largely  Ionized  Substances.— In  the  case  of  salts 
and  of  largely  ionized  acids  and  bases  the  ionization  values  derived 
from  conductance  or  from  freezing-point  measurements  do  not  change 
with  the  concentration  even  approximately  in   accordance  with  the 
mass-action  law.    Thus,  the  average  ionization  values  (7  obs.)  for  uni- 
univalent    salts   given   in   Art.    58,   the   "  ionization-constants "    (K) 


80  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

calculated  from  them,  and  the  ionization  values  (7  calc.)  calculated 
conversely  from  the  value  of  this  constant  at  0.1  normal  are  as  follows : 


Cone. 

0.001 

0.01 

0.02 

0.05 

0.10 

7Obs. 

0.96 

0.92 

0.90 

0.87 

0.84 

7  calc. 

0.998 

0.98 

0.96 

0.91 

0.84 

K 

0.023 

0.11 

0.16 

0.29 

0.44 

K/sJc 

(0.73) 

1.10 

1.13 

1.30 

1.39 

It  has  been  found  empirically  that  the  "  ionization-constant "  K 
varies  with  the  concentration  c  of  the  salt,  and  is  approximately 
proportional  to  the  square-root  of  that  concentration.  This  is  illus- 
trated by  the  comparatively  small  variations  of  the  values  of  JSYy/TT 
given  in  the  last  row  of  the  above  table.  This  principle  applies  also 
to  mixtures  of  salts,  the  "ionization-constant"  being  in  this  case 
dependent  on  the  sum  of  the  concentrations  (Sc)  of  all  the  salts  in 
the  mixture.  Thus,  for  a  salt  A+B~  the  principle  is  expressed  by  the 
equation : 

(A+)  X  (B-) 

— — — =K—  const.  X    V2c. 

\AB) 

ProT).  18.  a.  Show  that  this  principle  requires  that  in  a  mixture 
of  two  uniunivalent  salts  with  a  common-ion  (such  as  K+C1-  and 
Na+Cl~)  the  ionizations  of  the  two  salts  are  equal,  whatever  be  the 
relative  proportions  of  the  two  salts  in  the  mixture.  6.  Show  that 
the  principle  further  requires  that  the  ionization  y1  or  y2  of  either  salt 
in  the  mixture  is  equal  to  the  ionization  7  which  it  has  when  it  alone 
is  present  at  a  concentration  c  equal  to  the  sum  of  the  concentrations 

(°i  +  c«)  of  the  two  salts. 

Prob.  19.  What  are  the  concentrations  of  unionized  potassium 
chloride  and  of  unionized  sodium  chloride  in  a  mixture  0.02  normal 
in  KC1  and  0.08  normal  in  NaCl? 

The  reason  why  largely  ionized  substances  do  not  conform  to  the 
mass-action  law  is  not  known.  One  possible  explanation  of  this 
anomaly  is  that  the  mass-action  law  really  holds  true  in  the  case 
of  largely  ionized  substances,  but  that  the  conductance-ratio  is  not 
an  exact  measure  of  their  ionization.  This  would  imply  that  the 
mobility  of  the  ions  does  not  remain  constant  even  up  to  moderate 
concentrations,  as  was  assumed  in  Art.  55.  Another  explanation  is 
that  the  activity  either  of  the  ions  or  of  the  unionized  substance 
or  of  both  is  abnormal — in  other  words,  that  the  mass-action  effect  is 
not  proportional  to  the  concentration. 

The    simplest   way    of    accounting   fairly   satisfactorily    for    the 


BETWEEN  DISSOLVED  SUBSTANCES  81 

behavior  of  largely  ionized  substances  in  all  their  relations  seems 
at  the  present  time  to  be  the  adoption  of  the  following  hypotheses : 
(1)  the  conductance-ratio  is  a  substantially  correct  measure  of  their 
ionization;  (2)  the  ions  have  (approximately)  normal  activities;  and 
(3)  the  unionized  substance  has  an  activity  which  is  far  from  being 
proportional  to  its  concentration.  These  assumptions  will  therefore 
be  employed  in  this  book.  And  in  accordance  with  them,  in  dealing 
with  mass-action  problems,  the  equilibrium-equations  will  always  be 
so  written  as  to  involve  the  ions  rather  than  the  unionized  part  of 
largely  ionized  substances;  and  in  condition-equations  the  concen- 
trations of  the  unionized  parts  of  such  substances  will  be  calculated, 
not  from  their  ionization-constants,  but  from  the  ionization-values 
which  are  obtained  froin  the  conductance-ratio  or  from  the  average 
values  for  salts  of  the  same  valence  type. 

Prob.  20.  Formulate  the  equilibrium  equation  for  each  of  the  fol- 
lowing reactions  (taking  place  in  dilute  aqueous  solution/  : 

C12  +  H20  =  HC1  +  HC10;  and  Cla  -f  H2O  =  H+  -f  01-  -f  HC1O. 
Show  that  both  of  these  equilibrium  equations  cannot  hold  true,  since 
the  mass-action  law  does  not  apply  to  the  simple  ionization  of  largely 
ionized  acids. 

Pro  6.  21.  When  a  0.06  molal  solution  of  Cls  is  allowed  to  stand  at 
0°  till  equilibrium  is  reached,  31%  of  the  chlorine  is  converted  into 
hypochlorous  acid  and  hydrochloric  acid.  a.  Calculate  the  equilibrium- 
constant  of  the  reaction,  assuming  that  the  ionization  of  hydrochloric 
acid  is  95%.  6.  Calculate  what  the  initial  concentration  of  the  chlorine 
must  be  in  order  that  50%  of  it  may  be  converted  into  hypochlorous 
and  hydrochloric  acids  in  the  equilibrium  mixture. 

87.  The  Hydrolysis  of  Salts. — When  either  the  acid  or  base  of  a 
salt  has  a  very  small  ionization-constant,  the  salt  in  aqueous  solution 
reacts  with  the  water  to  an  appreciable  extent  with  formation  of  the 
acid  and  base.  Thus,  potassium  cyanide  in  0.01  normal  solution  at 
25°  is  decomposed  to  an  extent  of  3.5%  according  to  the  reaction 
K+CN-  +  H2O  =  K+OH-  +  HON.*  This  phenomenon  is  called 
hydrolysis;  and  the  fraction  of.  the  salt  hydrolyzed  is  called  the 
degree  of  hydrolysis,  or  simply  the  hydrolysis  (ft)-  The  equilibrium- 
constant  of  a  hydrolytic  reaction  is  called  the  hydrolysis-constant  of 
the  salt. 

*As  is  done  in  this  case,  it  is  convenient  to  indicate  largely  Ionized  sub- 
stances, whose  ionization  does  not  conform  to  the  mass-action  law.  by  attacbine 
-^  and  —  signs  to  their  ions,  and  .to  omit  such  signs  in  the  case  of  slightly 
ionized  substances,  whose  ionization  conforms  to  that  law. 


82  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

Prob.  22.  a.  Derive  a  mass-action  expression  showing  how  the 
hydrolysis  of  a  salt  like  potassium  cyanide  varies  with  the  concentra- 
tion of  the  salt.  b.  From  the  data  given  in  the  preceding  text  calculate 
the  degree  of  hydrolysis  and  the  concentration  of  free  base  in  a  0.1 
normal  KCN  solution  at  25°. — Assume  that  the  ionizations  of  the  salt 
and  free  base  are  equal. 

Prob.  28.  Derive  a  mass-action  expression  showing  how  the  hydrol- 
ysis of  a  salt  like  ammonium  cyanide,  of  which  the  acid  and  base  are 
both  slightly  ionized,  varies  with  the  concentration  and  ionization  of 
the  salt 

Pro 6.  24>  In  0.1  formal  solution  at  25°  ammonium  cyanide  is  40% 
hydrolyzed.  What  is  its  hydrolysis  in  0.01  formal  solution?  For  the 
ionization  values  refer  to  Art.  86. 

Prob.  25.  Show  how  the  hydrolysis-constants  of  salts  like  potas- 
sium cyanide  and  ammonium  cyanide  are  related  to  the  ionization- 
constants  of  water  (Kw),  of  the  acid  (J£A),  and  of  the  base  (^B). 

Prob.  26.  Calculate  the  hydrolysis  of  ammonium  acetate  in  0.1 
formal  solution  at  25°  from  the  ionization-constants  involved.  (The 
ionization-constants  of  acetic  acid  and  of  ammonium  hydroxide  have 
the  same  value  (1.8  X  10-6)  at  25°.) 

Prob.  27.  At  100°  the  ionization-constant  of  water  is  50  x  10-"; 
that  of  acetic  acid  is  1.1  x  10~5;  and  that  of  ammonium  hydroxide  is 
1.4  X  10~*.  Calculate  the  hydrolysis  of  ammonium  acetate  in  0.1  formal 
solution  at  100°,  and  compare  it  with  that  at  25°.  Assume  the  ioniza- 
tion of  the  salt  to  be  the  same  as  at  25°. 

Prob.  28. — Determination  of  Hydrolysis  by  Distribution  Experi- 
ments.— A  0.05  formal  solution  of  Na2NH4PO4  in  water  is  found  to  be 
in  equilibrium  with  a  0.00173  formal  solution  of  NH3  in  chloroform  at 
18°.  What  is  the  hydrolysis  of  the  salt?  The  distribution-ratio  for 
NH8  between  water  and  chloroform  at  18°  is  27.5. 

Prob.  29. — Determination  of  Hydrolysis  by  Reaction-Rate  Experi- 
ments.— The  specific  reaction-rate  at  100°  of  the  sugar  hydrolysis  has 
been  found  to  be  0.0386  in  a  solution  0.001  normal  in  HC1,  and  to  be 
0.0946  in  a  solution  0.01  formal  in  A1C18.  What  fraction  of  the  salt 
is  hydrolyzed  (into  Al(OH),  and  3HC1)? 

Prob.  80. — Determination  of  Hydrolysis  by  Conductance  Measure- 
ments.— The  specific  conductance  at  100°  of  a  0.025  formal  solution  of 
NH4Ac  (ammonium  acetate)  is  0.00685  reciprocal  ohms;  and  that  of  a 
solution  0.025  formal  in  NH4Ac  and  0.025  formal  in  NH4OH  is  0.00717 
reciprocal  ohms.  a.  Calculate  the  hydrolysis  of  the  ammonium  acetate 
in  the  first  solution,  assuming  that  in  the  second  solution  the  hydrol- 
ysis of  the  salt  and  the  conductance  of  the  base  are  both  negligible. 
b.  Calculate  the  hydrolysis  of  the  salt  in  the  second  solution,  c.  Calcu- 
late the  specific  conductance  of  the  free  base  in  the  second  solution. 
The  A0-value  for  NH4OH  at  100°  is  647. 


BETWEEN  DISSOLVED  SUBSTANCES  83 

88.  Displacement  of  One  Acid  or  Base  from  its  Salt  by  Another.— 
One  of  the  most  important  types  of  equilibrium  in  aqueous  solution 
is  the  partial  displacement  of  one  acid  or  base  from  its  salt  by  an- 
other ;  for  example,  that  of  acetic  acid  from  sodium  acetate  by  formic 
acid,  or  that  of  ammonium  hydroxide  from  ammonium  chloride  by 
sodium  hydroxide.  This  phenomenon  has  been  studied  experimentally 
by  the  methods  illustrated  by  Probs.  35-37.  Before  the  mass-action 
relations  involved  were  fully  understood,  the  extent  of  the  displace- 
ment was  taken  as  a  measure  of  the  relative  strengths  of  different 
acids  or  bases;  those  which  are  largely  displaced  from  their  salts 
being  called  weak  acids  or  bases,  and  those  which  cause  such  displace- 
ment being  called  strong  acids  or  bases.  It  is  shown  by  Probs.  31- 
33  that  the  mass-action  law  and  ionic  theory  give  a  comparatively 
simple  explanation  of  this  phenomenon  in  the  case  of  not  largely 
ionized  univalent  acids  or  bases;  also  that  the  relative  strengths  of 
different  acids  or  bases  are  determined  by  their  ionization,  weak  ones 
being  those  which  are  slightly  ionized  and  strong  ones  those  which 
are  largely  ionized. 

Proft.  31.  To  a  0.1  normal  solution  of  KNO2  is  added  at  25°  an 
equal  volume  of  a  0.1  normal  solution  of  acetic  acid.  a.  Calculate  the 
fraction  of  the  potassium  nitrite  that  is  converted  into  potassium  ace- 
tate. &.  Calculate  the  fraction  that  would  be  so  converted  if  the  acetic 
acid  solution  were  1.0  normal  (instead  of  0.1  normal). 

Pro 6.  82.  a.  For  the  general  case  expressed  by  the  equation 
B+A-  +  HA'  —  B+A'-  +  HA  where  the  solution  is  originally  c-forinal  in 
B+A-  and  cK-formal  in  HA',  derive  an  expression  for  the  fraction  x  of 
the  salt  B+A-  converted  into  the  salt  B+A'-,  assuming  that  the  acids  are 
not  largely  ionized.  6.  Derive  from  this  expression  the  relation  be- 
tween the  ionization-constants  of  the  acids  and  the  fractions  of  the  basic 
constituent  that  are  combined  with  the  two  acidic  constituents  for  the 
case  that  c  =  c' ;  and  state  the  principle  fully  in  words. 

Pro&.  33.  To  a  liter  of  0.1  normal  HC1  is  added  at  25°  a  liter  of 
0.1  normal  ammonia  (K  =  0.000018)  and  a  liter  of  0.1  normal  methyl- 
amine  (K  —  0.00050).  Calculate  the  fraction  of  the  acid  which  com- 
bines with  each  base. 

Prob.  34.  A  0.1  formal  solution  of  acetic  acid  is  added  to  an  equal 
volume  of  0.1  formal  NaHCO8  solution  at  25°,  the  carbon  dioxide  pro- 
duced being  kept  above  the  solution  at  a  pressure  of  1  atm.  Calculate 
the  concentrations  of  the  two  salts  and  two  acids  in  the  resulting  solu- 
tion (looking  up  the  data  needed).  Compare  this  result  with  that 
calculated  for  the  case  where  no  carbon  dioxide  is  allowed  to  escape 
from  the  solution. 


84  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

Pro&.  35. — Determination  of  Displacement  ~by  Volume  Measure- 
ments.— When  1000  g.  of  a  solution  containing  1KOH  is  mixed  with 
1000  g.  of  a  solution  containing  1CH8CO2H  there  is  an  increase  of 
volume  of  9.52  ccm.  When  the  former  solution  is  mixed  with  1000  g. 
of  a  solution  containing  1HCO2H  there  is  an  increase  of  volume  of 
12.39  ccm.  When  2000  g.  of  a  solution  containing  1HCO2K  is  mixed 
with  1000  g.  of  one  containing  1CH3CO2H  there  is  a  decrease  of  volume 
of  0.74  ccm.  Determine  what  fraction  of  the  formic  acid  is  displaced 
from  its  salt  by  the  acetic  acid.  Compare  this  result  with  that  calcu- 
lated from  the  ionization-constants. 

Note. — The  displacement  may  also  be  determined  experimentally 
by  measuring  the  quantities  of  heat  evolved,  instead  of  the  changes 
of  volume,  when  solutions  are  mixed  as  in  the  preceding  problem. 

Prol).  36. — Determination  of  Displacement  ly  Color  Measure- 
ments.— Paranitrophenol  is  a  slightly  ionized  acid  whose  solution  is 
colorless,  while  solutions  of  its  salts  have  a  yellow  color  which  in- 
creases proportionately  with  the  quantity  of  salt.  25  ccm.  of  a  0.01 
normal  solution  of  paranitrophenol  are  placed  in  each  of  two  tubes 
of  the  same  diameter ;  to  one  tube  are  added  25  ccm.  of  a  0.01  normal 
solution  'of  potassium  acetate;  and  to  the  other  tube  is  added  0.01 
normal  KOH  solution  until  on  looking  down  through  the  tubes  the 
colors  are  seen  to  be  the  same,  1.65  ccin.  of  the  KOH  solution  being 
required.  Calculate  the  fraction  of  the  paranitrophenol  which  exists 
in  the  first  mixture  in  the  form  of  its  salt;  and  calculate  the  ionization- 
constant  of  the  paranitrophenol. 

Prol.  37.  Suggest  a  method  (different  from  those  already  men- 
tioned) by  which  the  displacement  of  ammonium  hydroxide  from  am- 
monium chloride  by  sodium  hydroxide  might  be  determined. 

89.  Neutralization-Indicators. — An  acid  neutralization-indicator 
(such  as  litmus,  paranitrophenol,  or  phenol  phthalein)  is  a  mixture  of 
two  isomeric  acids*  (Hln'  and  HIn")  in  equilibrium  with  each  other, 
one  of  which  (HIn")  is  present  in  much  smaller  proportion  but  is 
so  much  more  ionized  than  the  other  (HIn')  that  when  a  base  BOH 
is  added  the  salt  produced  is  almost  wholly  of  the  form  B+In"~.  The 
substances  HIn'  and  B+In'~  are  different  in  color  from  the  substances 
HIn"  and  B+In"~,  the  color  being  determined  only  by  the  molecular 
structure  of  the  group  In.  It  follows  from  these  conditions  that  the 

*Thus  the  formulas  commonly  assigned  to  phenol  phthalein  and  its  potas- 
sium salt  are  : 


C8H4  —  C  and        rBH4  —  C 

CO   —    O    C«H*OH  COOK      X  CfiH4  =  ° 


BETWEEN  DISSOLVED  SUBSTANCES  85 

indicator  acid  changes  color  when  converted  into  its  salt.     It  will 

also  be  evident  from  Prob.  38  that  the  indicator  behaves  as  if  it  were 

a  single  acid  Hln  whose  salt  has  a  different  color  from  the  acid  itself. 

Prob.  38.     Show  by  formulating  the  equilibrium  equation  that  in 

the  case  of  an  indicator  such  as  has  been  described  — r=  const. 

(Hln') 

Note. — This  constant,  which  is  a  product  of  two  equilibrium-con- 
stants, will  be  called  the  indicator-constant  Klt 

Pro 6.  39.  Derive  a  relation  between  the  hydrogen-ion  concentration 
(H+),  the  indicator-constant  K^  jEnd  the  fraction  a?  of  the  indicator 
which  is  converted  into  the  colored  form  existing  in  alkaline  solution. 
(This  fraction  will  hereafter  be  called  simply  the  fraction  of  the  in- 
dicator transformed.) — In  indicator  problems  the  ionization  of  the  indi- 
cator salt,  as  well  as  that  of  any  other  salt  present,  may  be  regarded  as 
complete,  since  only  approximate  results  are  ordinarily  desired. 

The  relations  in  the  case  of  basic  indicators  (existing  almost 
wholly  in  the  two  differently  colored  forms  In'OH  and  In"+A~)  are 
entirely  similar  to  those  in  the  case  of  acid  indicators.  By  formulat- 
ing the  mass-action  equations,  it  can,  in  fact,  be  shown  that  the 
relation  derived  in  Prob.  39  between  the  hydrogen-ion  concentration 
and  the  fraction  (#)  of  the  indicator  transformed  holds  true  also  in  the 
case  of  basic  indicators.  By  adopting  as  the  value  of  the  indicator- 
constant  that  calculated  by  the  equation  K\  =  — a  basic  indi- 

1  —  x 

cator  may  therefore  be  treated  as  an  acid  indicator;    and  it  will  be 
so  treated  in  the  following  problems. 

Prob.  40. — Determination  of  the  Indicator-Constant. — To  100  ccrn. 
of  a  solution  0.1  normal  in  K>A<r  and  0.01  normal  in  HAc  are  added 
10  ccm.  of  a  0.01  normal  solution  of  paranitrophenol.  This  solution  is 
found  to  have  the  same  color  as  a  solution  made  by  adding  0.50  ccm. 
of  0.01  normal  paranitrophenol  solution  and  9.5  ccm.  of  0.01  normal 
KOH  solution  to  100  ccm.  of  water,  a.  Calculate  the  indicator-constant, 
neglecting  the  quantity  of  the  acetic  acid  displaced  from  its  salt  by  the 
small  proportion  of  paranitrophenol  present.  b.  Show  what  percentage 
error  is  made  in  (H+),  and  therefore  in  Elf  by  neglecting  this  displaced 
quantity. 

Pro  b.  41.  Determination  of  Small  Hydrogen-Ion  Concentrations  by 
Means  of  Indicators. — When  a  small  proportion  of  phenol  phthalein 
(TTj  — 10-10)  is  added  to  a  0.1  formal  solution  of  NaHCO3  at  25°  the 
indicator  is  found  by  color  comparisons  to  be  6.0%  transformed  into  its 
salt.  Calculate  the  hydrogen-ion  and  hydroxide-ion  concentration  in  the 
solution. 


86  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

Titration  of  Acids  and  Bases.— 

Prob.  42.  100  ccm.  of  a  0.2  normal  solution  of  an  acid  whose  ion- 
ization-constant  is  10-6  are  titrated  at  25°  (where  the  ionization-con- 
stant  of  water  Kw  is  10-")  with  0.2  normal  KOH  solution.  Calculate 
the  hydrogen-ion  concentration  (H+)  in  the  mixture  when  99.0,  99.5, 
99.8,  100.0,  100.2,  100.5,  and  101.0  ccm.  of  the  KOH  solution  have  been 
added. 

Prob.  JtS.    The  values  of  (H+)  obtained  in  Prob.  42  and  those  cal- 
culated  in   the   same   way   for    acids   whose    ionization-constants   (KA) 
are  10~3,  10-7,  and  10-8  for  different  ratios  (B/A)  of  the  quantity  of  0.2 
normal  KOH  to  the  quantity  of  0.2  normal  acid  are  at  25°  as  follows : 
Ratio  Values   of   the   hydrogen-ion   concentration  for 

B/A  KA  =  1(T3  K^  —  10-5  KA  =  10-1  KA  =  10-» 

0.9SO  2  X  10-6  2  X  10~7  2  X  10-9  2.4  X  10-" 

0.990  IxlO-5  IXlO-7  1X10-9  1.6X10-" 

0.995  5  X  1°"6  5  X  1°~8  5  X  10-10  1.2  X  1°~" 

0.998  2  X  10-'  2  x  10'8  2.4  X  lO"10  1.1  X  10~" 

1.000  1  X  1°~8  !  X  10-'  1  X  1°"10  1-0  X  10~n 

1.002  5  X  10-u  5  X  10-11  4  X  10-11  0.9  X  10"11 

1.005  2  X  10-11  2  X  10-u  2  X  10-"  0.8  X  10-" 

1.010       i  x  lo-11      i  x  io-u      i  x  io-u        0.6  x  io-n 

1.020          5x10-"         5X10-"         5x10-"  0.4x10'" 

These  numbers  are  also  the  values  of  (OH-)  for  the  case  that  a  0.2 
normal  solution  of  a  base  having  an  ionization-constant  equal  to  10~s, 
10~6,  10~T,  or  10-9  is  titrated  with  a  0.2  normal  solution  of  a  strong  acid 
(like  HC1),  provided  the  numbers  in  the  first  column  denote  the  ratio 
(A/B)  of  the  quantity  of  acid  to  the  quantity  of  base.  (Thus,  when 
the  ratio  A/B  is  0.98,  the  value  of  (OH-)  is  2  x  10'6  for  a  base  for 
which  KB  — 10-8.) 

a.  Plot  the  common  logarithms  of  these  values  of  (H+)  as  ordi- 
nates  against  the  corresponding  values  of  the  ratio  of  base  to  acid  as 
abscissas,  for  each  of  the  four  acids.  b.  On  the  right-hand  side  of  the 
same  diagram  write  in  a  scale  of  values  of  log10(OH-)  corresponding 
to  the  values  of  log,0(H+)  on  the  left-hand  side;  and  at  the  top  of  the 
diagram  write  in  a  scale  of  ratios  of  acid  to  base  (A/B)  corresponding 
to  the  scale  of  ratios  of  base  to  acid  at  the  bottom  (thus,  B/A  =  0.98 
corresponds  to  A/B  =  1.02).  Now  make  plots  on  the  same  diagram 
showing  how  (H+)  or  (OH~)  varies  in  titrating  bases  of  ionization- 
constants  10-s,  10-5,  10-7,  and  10-8  with  HC1  at  25°,  similar  in  all  respects 
to  the  plots  previously  made  for  the  four  acids. 

Prob.  )t!t.  a.  From  a  study  of  the  diagram  of  Prob.  43  tabulate 
the  values  between  which  the  indicator-constant  must  lie  in  order  that 
the  titration  of  each  of  the  four  acids  and  four  bases  may  be  correct 
\vithin  0.2%,  assuming  that  the  indicator  is  9%  transformed.  6.  Show 
from  the  plot  what  percentage  error  would  be  made  in  using  phenol 


INVOLVING  SOLID  SUBSTANCES  87 

phthalein  (^  =:10-10)  in  titrating  an  acid  for  which  KA  =  1Q-B  when 
the  fraction  of  the  indicator  transformed  is  1%,  9%,  and  50%;  also 
in  titrating  an  acid  for  which  K^  — 10~7  when  the  fraction  of  the  in- 
dicator transformed  is  9%  and  50%.  c.  If  the  acid  for  which 
K  =10-9  were  titrated  with  the  aid  of  an  indicator  for  which 
KA=  O-12  (which  is  not  far  from  the  value  for  trinitrobenzene),  what 
error  would  be  made  when  the  fraction  of  the  indicator  transformed 
is  5%,  9%,  and  15%  transformed?  (Note  that  in  a  titration  carried 
out  in  the  usual  way  the  fraction  transformed  is  not  determined  more 
closely  than  this.  Note  also  that  an  error  in  the  assumed  value  of  the 
indicator-constant  would  affect  the  results  in  the  same  way  as  a  vari 
ation  in  the  fraction  transformed.) 

Prob.  45.  Calculate  the  value  of  (H+)  at  the  end-point  in  titrating 
with  phenol  phthalein  (.5^  — 10-10)  when  the  fraction  of  it  trans- 
formed x  is  5%  and  20%  ;  with  rosolic  acid  (J£j  =  10-8)  when  x  =  5% 
and  20%;  with  paranitrophenol  (KT  =  10-7)  when  x  =  l%  and  20%; 
and  with  methyl  orange  (Kt  —  5  X 10^)  when  x  —  80%  and  95% 
(these  being  about  the  limits  practicable  in  a  titration).  Draw  in  on 
the  diagram  made  in  Prob.  43  horizontal  lines  representing  these  limit- 
ing values  of  (H+)  for  the  four  indicators.  Letter  the  curves  and  lines 
on  the  diagram  so  as  to  show  what  each  represents. 

Prob.  46.  With  the  aid  of  the  diagram  show  which  of  these  in- 
dicators would  give  a  result  accurate  within  0.2-0.3%  in  titrating 
a,  NH4OH  with  HC1 ;  6,  HNO2  with  KOH;  c,  aniline  (KB  —  4  X  10'10) 
with  HC1. 


REACTIONS    INVOLVING    SOLID    SUBSTANCES 

90.  Form  of  the  Mass-Action  Expression. — When  a  substance 
present  as  a  solid  phase  is  involved  in  a  reaction  with  gaseous  sub- 
stances at  small  pressures,  it  has  in  the  gaseous  phase  of  all  equi- 
librium mixtures  at  any  definite  temperature  the  same  pressure, 
namely,  one  equal  to  the  vapor-pressure  of  the  solid  substance.  Simi- 
larly, when  a  substance  present  as  a  solid  phase  is  involved  in  a 
reaction  with  dissolved  substances  at  small  concentrations,  it  has 
in  the  liquid  phase  of  all  equilibrium  mixtures  at  any  definite  tem- 
perature, the  same  concentration,  namely,  one  equal  to  its  concen- 
tration in  a  solution  saturated  with  the  solid  substance  and  con- 
taining no  other  solutes.  Hence  the  pressure  in  the  gaseous  phase 
or  the  concentration  in  the  liquid  phase  of  any  substances  which  are 
also  present  as  solid  phases  may  be  left  out  in  formulating  the  mass- 


88  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

action  expression;  their  constant  pressures  or  concentrations  being 
understood  to  be  included  in  the  equilibrium-constant.  Thus,  the 
mass-action  expression  for  the  reaction*  Fe  -f-  H20  =  FeO  -f-  H2  is 
simply  pHi/PH20=-K. 

91.  Reactions  involving  Solid  and  Gaseous  Substances.— The 
simplest  type  of  the  reactions  involving  solid  and  gaseous  substances 
is  that  in  which  only  one  substance  is  present  in  appreciable  quan- 
tity in  the  gaseous  phase.  Examples  of  this  type  are:  2Ag,0  = 
4Ag  +  Oa;  CaCOs  =  CaO-t-Cfat;  and  CaS04.2H20  =  CaSO,+  2#20. 
The  mass-action  expression  for  this  case  is  simply  p  =  K,  which  sig- 
nifies that  at  any  definite  temperature  there  is  only  one  pressure  of 
the  gas  at  which  there  can  be  equilibrium.  This  pressure  is  called 
the  dissociation-pressure  of  the  substance  undergoing  decomposition 
(thus  of  the  silver  oxide,  the  calcium  carbonate,  or  the  gypsum). 
If  the  pressure  of  the  gas  is  kept  larger  than  this  pressure,  the  reaction 
takes  place  completely  in  one  direction,  with  the  result  that  the  gas 
is  entirely  absorbed;  and  if  the  pressure  is  kept  smaller,  the  reaction 
takes  place  completely  in  the  other  direction,  with  the  result  that  the 
dissociating  substance  completely  decomposes.  This  important  char- 
acteristic of  reactions  of  this  type  will  be  fully  considered  in  the 
next  chapter. 

Other  types  of  reactions  involving  solid  and  gaseous  substances 
are  illustrated  by  Probs.  47-51. 

Prol).  47.  When  solid  NH4SH  is  placed  in  a  vacuous  space  at  25° 
n  pressure  of  500  nun.  is  developed,  owing  to  the  dissociation  of  the 
snlt.  which  is  itself  not  appreciably  volatile,  into  NH3  and  H2S.  What 
increase  of  pressure  would  take  place  if  solid  NH4SH  were  introduced 
into  a  space  which  already  contained  hydrogen  sulphide  at  a  pressure 
of  300  nun.? 

I'rob.  J,8.  The  equilibrium-constant  (expressed  in  atmospheres)  of 
the  reaction  2NaHC08  —  Na2C08  J-  CO2  -f  J/,O  at  100°  is  0.23.  A  cur- 
rent of  moist  carbon  dioxide  is  passed  at  1  atm.  over  solid  sodium 
hydrogen  carbonate  (in  order  to  free  it  from  adhering  water).  TTow 
great  must  the  niol-fraction  of  the  water  in  the  gas  be  to  prevent  de- 
composition of  the  salt? 

I'rob.  1,9.  A  current  of  air  at  1  atm.  is  passed  over  carbon  at 
1000°.  What  will  be  the  molal  ratio  of  CO,  to  CO  in  the  issuing  gas, 

*Solid  substances  occurring  in  chemical  equations  are  represented  by  black 
letters  and  jraseons  substances  by  italics,  whenever  it  is  important  to  indicate  the 
state  of  aggregation. 


INVOLVING  SOLID  SUBSTANCES  89 

assuming  equilibrium  to  be  reached?  The  equilibrium-constant  for  the 
reaction  C  +  CO2  =  2CO  at  1000°  is  140. 

Pro&.  50.  At  1120°  the  gaseous  mixture  in  equilibrium  with  iron 
and  ferrous  oxide  consists  of  54  mol-percent  H2  and  46  mol  -percent 
H2O.  Calculate  the  dissociation-pressure  of  ferrous  oxide.  (Refer  to 
Probs.  8  and  9.) 

Pro 6.  51.  In  the  blast-furnace  process  iron  is  reduced  by  the  reac- 
tion FeO  -f  CO  =  Fe  -f  CO¥  Calculate  the  least  quantity  of  carbon 
monoxide  that  could  reduce  one  formula  weight  of  FeO  at  1120°,  using 
the  dissociation-pressure  of  ferrous  oxide  found  in  Prob.  50. 

92.  Reactions  involving  Solid  and  Dissolved  Substances. 
Prob.  52.    At  25°  the  solubility  of  iodine  in  water  is  0.0013  molal ; 

and  its  solubility  in  0.1  normal  KI  solution  is  0.0517  molal,  the  increase 
being  due  to  the  reaction  K+I-  +  Ij  =  K>I,-.  Calculate  its  solubility  in 
0.01  normal  KI  solution  at  25°. 

Pro  ft.  53.  Hypochlorous  acid  is  prepared  by  the  action  of  chlorine 
solution  on  solid  mercuric  oxide.  What  is  the  expression  for  the  equilib- 
rium of  the  reaction? 

93.  Solubility   Effects    in   the    Case   of   Largely   Ionized    Sub- 
stances.—  The  mass-action  law  evidently  requires,  in  all  dilute  solu- 
tions saturated  with  the  same  solid  substance,  that  the  concentration 
of  the  unionized  substance  have  the  same  value,  and  also  that  the 
product  of  the  concentrations  of  its  ions  have  the  same  value,  what- 
ever other  substances  may  be  present   (at  small  concentrations)   in 
the  solution.     Owing,  however,  to  the  considerable  deviations  from 
the  mass-action  law   (referred  to  in  Art.  86)   which  largely  ionized 
substances   exhibit   even   at   moderately   small   concentrations,    it    is 
clear  that  one  or  both  of  these  principles  must  be  inexact  when  ap- 
plied to  such  substances.     Experiment  has  shown  that,  in  tbis  case 
as  in  others,  a  fairly  good  agreement  with  the  facts  is  secured  (up  to 
about  0.2  normal)  by  employing  tbe  principle  involving  the  ions — 
that  expressing  tbe  constancy  of  the  ion-concentration  product. 

In  solutions  saturated  witb  tri-ionic  substances,  such  as  silver 
sulphate  or  calcium  hydroxide,  the  relations  are  further  complicated, 
except  in  very  dilute  solutions,  by  the  probable  presence  in  consid- 
erable proportion  of  intermediate  ions,  such  as  AgSO4~  or  CaOH*. 
Owing  to  lack  of  knowledge  of  the  proportion  in  which  such  ions  are 
present,  the  principle  of  the  constancy  of  tbe  ion-concentration 
product  (for  example,  of  the  products  (Ag+)a  X  (SO4=)  and  (Oai+)  X 
(OH")a),  combined  with  tbe  assumption  that  the  conductance-ratio 


90  EQUILIBRIUM  OF  CHEMICAL  REACTIONS 

gives  the  ionization  of  the  salt  into  the  simple  ions,  can  be  used  with 
fairly  satisfactory  results  only  when  the  solubility  of  the  solid  sub- 
stance and  the  concentration  of  any  other  substance  present  are  both 
very  small.* 

Solubility-Decrease  by  Substances  with  a  Common-Ion. — 

Pro b.  54'  0"  Derive  an  expression  for  the  solubility  s  of  silver 
chloride  in  a  dilute  sodium  chloride  solution  of  concentration  c  in 
terms  of  its  solubility  s0  in  water,  assuming  the  salts  to  be  completely 
ionized,  b.  Calculate  the  ratio  s/s0  for  c  =  s0,  fo~  c  =  2s0,  and  for 
c  =  106v 

Prob.  55.  The  solubility  at  25°  of  thallous  chloride  (T1C1)  is 
0.0161  normal.  Calculate  its  solubility  in  a  0.05  normal  solution  of 
potassium  chloride.  Estimate  the  ionization-values  involved  with  the 
aid  of  the  average  values  for  uniunivalent  salts  given  in  Art.  S6. 
Compare  the  calculated  solubility  with  the  value  (0.00592  normal) 
found  by  experiment. 

Prob.  56.  The  solubility  of  magnesium  hydroxide  in  water  at  1S° 
is  1.4  x  10-'  formal,  a.  Calculate  its  solubility  in  0.002  formal  NaOLJ 
solution.  6.  Calculate  its  solubility  in  0.001  formal  MgCl2  solution. 
(Assume  that  the  substances  are  completely  ionized.) 

Prob.  57. — Solubility-Increase  through  Complex-Formation. — The 
solubility  of  silver  chloride  in  water  at  25°  is  1.1  x  10~B  formal.  Cal- 
culate its  solubility  in  0.1  formal  NH3  solution.  There  is  formed  a 
complex  ion  by  the  reaction  Ag+  -|-2NH3  =  Ag(NH8)2+,  its  equilibrium- 
constant  (commonly  called  the  complex-constant)  having  the  value 
1.4  x  107.  Assume  the  ionization  of  the  salts  to  be  complete,  and  make 
any  other  justifiable  simplification. 

Prob.  58. — Solubility-Increase  through  Metathesis. — The  solubil- 
ity of  magnesium  hydroxide  in  water  at  18°  is  1.4  x  10~4  formal.  Cal- 
culate its  solubility  in  0.002  formal  NH4C1  solution.  Assume  that  the 
salts  and  the  magnesium  hydroxide  are  completely  ionized.  Neglect 
the  concentration  of  OH-  in  comparison  with  that  of  NH4OH. 

Conversion  of  One  Solid  Substance  into  Another. — 

Prob.  59.  The  solubility  of  silver  thiocyanate  is  1.2  x  iO'6  formal 
and  that  of  silver  bromide  is  0.6  x  10-'  formal  at  25°.  a.  Calculate  the 
equilibrium-constant  of  the  reaction  AgSCN  -f  K+Br~  =  AgBr  -j-  K>SCN- 

*How  small  these  concentrations  must  be  in  order  that  the  solubility  in 
the  presence  of  an  added  salt  may  be  calculated  from  that  in  water  with  an 
accuracy  of  a  few  percent  depends  on  the  valence  type  of  the  added  substance. 
When  this  substance  has  the  same  univalent  ion  as  the  substance  saturating 
the  solution  (e.  g.,  when  Ag+NO3-  is  added  to  Ag+2SO4=,  or  Na+OH-  to 
Ca+-MOH-)2),  the  total  concentration  may  be  as  high  as  0.05  normal;  but  when 
the  added  substance  has  the  same  bivalent  ion  (e.  g.,  when  K2+SO4=  is  added  to 
Ag+2SO4=  or  Ca++(NO3-)2  to  Ca+J(OH-)2)  the  total  concentration  must  not  be 
greater  than  0.002  normal. 


INVOLVING  SOLID  SUBSTANCES  91 

in  dilute  solution.  6.  If  8.3  g.  of  solid  silver  thiocyanate  are  treated 
with  200  cc.  0.1  formal  KBr  solution,  what  proportion  of  the  silver 
salt  is  converted  into  bromide?  c.  What  volume  of  the  0.1  formal 
KBr  solution  would  convert  the  silver  thiocyanate  completely  into 
bromide?  d.  With  what  solutions  of  potassium  thiocyanate  and  potas- 
sium bromide  could  the  silver  thiocyanate  be  treated  without  any 
change  taking  place? 

Pro 6.  60.  Determine  the  ratio  of  carbonate  to  hydroxide  in  the 
solution  obtained  by  digesting  at  25°  a  0.1  formal  Na,CO,  solution  with 
excess  of  solid  Ca(OH)a  (as  in  the  technical  process  of  causticizing 
soda).  The  solubility  of  calcium  hydroxide  is  0.020  formal,  and  the 
product  (Ca~)  x  (CO,=)  has  the  value  3  x  10~9  in  water  saturated 
with  calcium  carbonate.  Assume  that  the  substances  are  completely 
ionized. 


CHAPTER   VII 

EQUILIBRIUM  OF  CHEMICAL  SYSTEMS  IN  RELATION 
TO  THE   CHARACTER  OF  THE   PHASES   PRESENT 


94.  Fundamental  Conceptions. — In  this  chapter  are  considered  the 
principles  relating  to  the  number,  state  of  aggregation,  and  composi- 
tion of  the  phases  (defined  as  in  Art.  23)  which  coexist  in  equilibrium 
with  one  another  when  a  given  substance  is  subjected,  or  when  mix- 
tures in  various  proportions  of  two  or  more  given  substances  are 
subjected,  to  different  temperatures  and  pressures. 

The  kinds  of  phenomena  to  be  considered  are  illustrated  by  the 
following  examples.  The  state  in  which  the  substance  water  exists  is 
determined  by  the  temperature  and  pressure — thus,  whether  it  exists 
in  the  form  of  a  single  phase  as  ice,  as  liquid  water,  or  as  vapor;  in 
the  form  of  two  phases  as  ice  and  liquid  water,  as  ice  and  vapor,  or 
as  liquid  water  and  vapor;  or  in  the  form  of  the  three  phases,  ice, 
liquid  water,  and  vapor.  So  also  in  the  case  of  two  substances,  such 
as  carbon  bisulphide  and  acetone,  there  are,  as  shown  by  the  vapor- 
pressure-composition  and  boiling-point-composition  diagrams  of 
Arts.  32  and  33,  definite  conditions  of  pressure  and  temperature  at 
which  any  definite  mixture  of  the  two  substances  forms  two  phases — 
a  liquid  phase  and  a  vapor  phase;  and  under  these  conditions  the 
composition  of  each  phase  is  also  definite. 

Certain  fundamental  conceptions  may  first  be  presented. 

The  combination  of  matter  under  consideration  is  called  the  system. 
A  system  is  therefore  defined  when  the  nature  and  quantity  of  the 
substances  of  which  it  is  composed  are  specified.  Throughout  this  chap- 
ter are  to  be  considered  the  conditions  which  determine  the  state  of 
systems  in  equilibrium  composed  of  the  same  kinds  of  substances  in 
various  proportions;  that  is,  of  a  series  of  systems  having  the  same 
qualitative,  but  varying  quantitative  composition. 

Any  pure  substances  (Art.  2)  which,  put  together  in  suitable  propor- 
tions, will  produce  each  and  every  phase  with  such  varying  composition 
as  it  may  have  in  the  systems  under  consideration  are  called  the 
components,  the  pure  substances  being  so  chosen  that  the  systems  can 
be  produced  out  of  the  smallest  number  of  them. 

92 


FUNDAMENTAL  CONCEPTIONS  93 

Any  phase  may  be  produced  out  of  the  components  either  directly 
or  as  a  result  of  equilibria  established  between  them.  For  example,  a 
gaseous  phase  containing  hydrogen,  oxygen,  and  water-vapor  at  high 
temperatures  where  chemical  equilibrium  is  established  between  these 
substances  can  be  produced  by  putting  together  only  hydrogen  and 
oxygen;  and  the  phase  is  therefore  said  to  consist  of  these  two  com- 
ponents. Such  a  gaseous  phase  at  low  temperatures,  where  no  chemical 
reaction  takes  place  between  the  hydrogen  and  oxygen,  can  be  produced 
only  by  putting  together  with  these  two  substances  also  water ;  so  that 
in  this  case  the  phase  is  said  to  consist  of  these  three  components.  It 
will  be  clear  from  this  example  that  the  number  of  components  is 
determined  not  only  by  the  substances  present,  but  also  by  the  equi- 
libria which  are  established  between  them. 

Another  aspect  of  the  matter  may  be  considered.  In  a  gaseous 
phase  where  hydrogen,  oxygen,  and  water  are  in  chemical  equilibrium, 
one  particular  system  could  be  produced  by  taking  only  the  one 
substance  water;  but  the  phase  considerations  of  this  chapter  have 
reference  always  to  a  series  of  systems  of  the  same  qualitative,  but 
varying  quantitative  composition;  and  such  a  series  containing 
hydrogen,  oxygen,  and  water-vapor  in  every  possible  proportion  cannot 
be  produced  out  of  water  alone,  but  can  be  produced  out  of  hydrogen 
and  oxygen.  As  another  example  consider  a  gaseous  phase  containing 
HC1,  O2,  C12,  and  H,0.  By  taking  HC1  and  O2  as  two  components, 
the  other  two  substances  can  be  produced  out  of  them  by  the  reaction 
O,  -h  4HC1  =  2C12  +  2H2O,  but  only  in  equivalent  proportions.  To 
produce  a  system  containing  the  four  substances  in  any  proportion 
whatsoever,  it  is  necessary  to  make  use  of  a  third  component,  either 
Cl,  or  H2O.  Thus,  either  by  adding  Cl,  to  a  mixture  containing  O2 
and  HC1  in  any  proportions  and  C12  and  H2O  in  equivalent  propor- 
tions, or  by  removing  C12  from  such  a  mixture,  any  composition  what- 
ever can  evidently  be  secured.  The  systems  are  therefore  said  to 
consist  of  three  components.  In  the  above  description  O2,  HC1,  and 
C12  have  been  used  as  the  components;  but  evidently  any  other  three 
of  the  four  substances  might  be  equally  well  employed.  It  is,  how- 
ever, preferable  to  select  as  components  substances  of  as  simple  a 
composition  as  possible;  thus  in  this  case,  O2,  HC1,  and  C12  rather 
than  O2,  HC1,  and  H2O. 


94  PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 

Profc.  1.  With  the  aid  of  the  preceding  considerations,  specify  the 
components  which  will  produce  phases  containing  the  following  chemical 
substances,  assuming  chemical  equilibrium  to  be  established  between 
them.  In  answering  this  question,  write  chemical  equations  showing 
how  all  the  chemical  substances  can  be  produced  from  the  pure  substances 
used  as  components,  a.  Gaseous  H2,  O2,  CO,  CO2,  H2O.  6.  Solution  contain- 
ing H2O,  (H2O)2,  NaC12H2O,  NaCl,  Na+,  C1-.  c.  Solution  containing 
H2O,  NH4CN,  NH4%  CN~,  NH8,  NH4OH,  HCN. 

.Proft.  2.  Specify  the  components  of  the  systems  that  exist  in  the 
following  groups  of  phases,  assuming  chemical  equilibrium  to  be  estab- 
lished between  the  substances  named :  a.  Solid  NH4C1,  gaseous  NH8  and 
HC1.  6.  Solid  CaCO8,  solid  CaO,  gaseous  CO2.  c.  Solid  MgSO4.7H2O, 
solid  MgSO4.6H2O,  solution  of  Mg+^SCV  in  water,  water- vapor,  d.  Solid 
carbon ;  gaseous  H2O,  H2,  CO,  and  CO2.  e.  Solid  iron,  solid  FeO,  gaseous 
CO  and  CO2.  /.  Solid  iron,  solid  FeO,  solid  carbon,  gaseous  CO  and  CO2. 

It  is  a  fundamental  law  of  the  equilibrium  between  phases  that  the 
absolute  quantity  of  the  different  phases  does  not  influence  their  com- 
position. Thus  the  composition  of  a  solution  in  equilibrium  with  a 
solid  salt  is  not  dependent  on  the  quantity  of  the  solid  in  contact  with 
the  solution.  The  composition  of  the  vapor  in  equilibrium  with  a 
definite  liquid  solution  is  not  dependent  on  the  quantities  of  liquid 
and  vapor  in  contact  with  one  another.  The  proportion  of  hydrogen 
and  water-vapor  in  equilibrium  with  solid  iron  and  solid  ferrous  oxide 
is  not  dependent  on  the  quantities  of  these  solid  phases  in  contact 
with  the  gaseous  phase.  The  rate  at  which  equilibrium  is  established 
between  phases  is,  however,  greatly  increased  by  increasing  the  extent 
of  the  surfaces  between  them. 

A  system  is  determined  by  its  composition,  that  is,  by  the  quanti- 
ties of  its  components,  as  described  in  the  preceding  paragraphs.  The 
state  of  a  system  is  determined  when  the  nature  and  quantity  of  each 
of  its  different  phases  are  specified  and  when  the  specific  properties  of 
each  phase,  such  as  its  density,  specific  conductance,  index  of  refrac- 
tion, etc.,  have  definite  values.  In  order  to  determine  fully  the  specific 
properties  of  any  one  phase  of  a  system,  it  is  necessary  to  specify,  in 
addition  to  the  proportions  of  its  components,  any  external  factors 
which  affect  these  properties.  The  only  external  factors  which  com- 
monly have  an  appreciable  influence  are  the  pressure  and  temperature ; 
and,  in  the  following  considerations  relating  to  the  equilibrium  of 
phases,  these  factors  alone  are  taken  into  account,  and  their  values 
are  assumed  to  be  uniform  throughout  all  the  phases  of  a  system. 


ONE-COMPONENT  SYSTEMS 


95 


Briefly  stated,  the  purpose  of  this  chapter  is  to  show  how  the  nature, 
the  quantity,  and  the  composition  of  the  phases  of  a  system  may  be 
represented  when  the  composition  of  the  system  as  a  whole  and  the 
pressure  and  temperature  are  given. 

Systems  are  classified  according  to  the  number  of  their  components. 
In  this  chapter  one-component  systems  are  first  considered;  then  a 
general  principle,  known  as  the  phase  rule,  applicable  to  systems  with 
any  number  of  components,  is  presented;  and  finally  two-component 
and  three-component  systems  are  discussed. 

ONE-COMPONENT  SYSTEMS 

95.  Representation  of  the  Equilibrium-Conditions  by  Diagrams. — 
In  the  case  of  one-component  systems  the  conditions  under  which 
the  different  phases  exist  in  equilibrium  with  each  other  are  con- 
veniently represented  by  pressure-temperature  diagrams;  for  the 
state  of  any  phase  of  such  a  system  is  evidently  determined  when  the 
pressure  and  temperature  are  specified. 


0.05  , 


0.04 


0.03- 


0.02 


0.01 


RHOMS/G 


H 


MO/VOCL  //V/&          „  7_ 

u— _--Vc 

/G 


90 


95 


100 


105  110 

FIGURE  7 


115 


120 


125 


Figure  7  shows  a  part  of  the  temperature-pressure  diagram  for 
the  component  sulphur,  which  forms  not  only  liquid  and  gaseous 
phases,  but  also  two  solid  phases,  known  from  their  crystalline  forms 


96  PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 

as  rhombic  and  monoclinic  sulphur.  In  the  diagram  the  (vapor- 
pressure)  curve  AB  represents  the  pressures  at  which  rhombic  sul- 
phur and  sulphur-vapor  are  in  equilibrium  at  various  temperatures; 
the  (vapor-pressure)  curve  BC  represents  the  pressures  at  which 
monoclinic  sulphur  and  sulphur- vapor  are  in  equilibrium  at  various 
temperatures;  and  the  (transition-temperature)  curve  BE  represents 
the  temperatures  at  which  monoclinic  and  rhombic  sulphur  are  in 
equilibrium  at  various  pressures,  Temperatures,  like  these,  at  which 
two  solid  phases  are  in  equilibrium  with  each  other  are  called  transi- 
tion-temperatures. 

The  point  of  intersection  B  of  these  three  curves  shows  the  only 
temperature  and  pressure  at  which  rhombic  sulphur,  monoclinic 
sulphur,  and  sulphur-vapor  are  in  equilibrium  with  one  another.  A 
point,  like  this,  at  which  three  phases  coexist  is  called  a  triple  point. 

The  (vapor-pressure)  curve  CD  represents  the  pressures  at  which 
liquid  sulphur  and  sulphur-vapor  are  in  equilibrium  with  each  other 
at  various  temperatures;  and  the  (melting-point)  curve  OF  repre- 
sents the  temperatures  at  which  monoclinic  sulphur  and  liquid 
sulphur  are  in  equilibrium  at  various  pressures.  The  point  C  is  evi- 
dently a  second  triple  point  at  which  monoclinic  sulphur,  liquid 
sulphur,  and  sulphur-vapor  co-exist.  As  indicated  on  the  diagram, 
the  fields  between  the  different  lines  show  the  conditions  under 
which  the  sulphur  exists  as  a  single  phase. 

Profc.  3.  Describe  with  the  aid  of  the  diagram  the  changes  that 
take  place,  a,  when  sulphur  is  heated  in  an  evacuated  tube  in  contact 
with  its  vapor  from  90°  to  125°;  6,  when  sulphur  is  allowed  to  cool 
from  125°  to  90°,  the  pressure  being  kept  constant  at  0.04  mm. 

The  curves  BE  and  OF  in  Figure  7  are  very  nearly  vertical  lines ; 
for  increase  of  pressure  always  produces  a  relatively  small  change 
in  the  transition  or  melting  temperature.  Thus  the  transition-tem- 
perature of  the  two  solid  forms  of  sulphur  increases  about  0.04°, 
and  the  melting-point  of  monoclinic  sulphur  increases  about  0.03°, 
per  atmosphere  of  pressure.  With  certain  substances  the  effect  of 
pressure  is  to  decrease  the  transition  or  melting  temperature;  thus 
the  melting-point  of  ice  is  lowered  by  0.0076°  by  an  increase  of  pres- 
sure of  one  atmosphere. 


ONE-COMPONENT  SYSTEMS  97 

Prob.  4.  At  what  temperature  and  pressure  is  there  a  triple  point 
in  the  system  composed  of  water?  The  freezing-point  at  1  atm.  is  0°. 
For  the  other  data  needed  refer  to  Art.  34  and  the  preceding  text. 


96.   Unstable  Forms. 

Prob.  5.  a.  To  what  equilibria  do  the  curves  BG,  GO,  and  GH  and 
the  point  G  in  Figure  7  correspond?  b.  Considered  with  reference  to 
these  equilibria,  to  what  form  of  sulphur  do  the  fields  EBGH,  HGCF, 
and  GBC  correspond?  c.  In  what  sense  are  these  equilibria  unstable? 

Prob.  6.  Draw  a  sulphur  diagram  extending  to  pressures  above  the 
triple-point  between  rhombic,  monoclinic,  and  liquid  sulphur,  which  lies 
at  151°  and  1280  atm. 

Prob.  7.  It  will  be  noted  that  the  unstable  form  at  any  temper- 
ature has  the  greater  vapor-pressure.  Prove  that  this  must  be  so  by 
showing  what  would  happen  if  the  two  forms  were  placed  beside  each 
other  in  an  evacuated  apparatus. 

Prob.  8.  Prove  by  a  similar  consideration  that  the  unstable  form 
must  also  have  the  greater  solubility  in  any  solvent,  such  as  carbon 
bisulphide.  (Note  that  a  substance  is  commonly  present  in  the  same 
molecular  form  in  the  solutions  produced  by  dissolving  its  different  solid 
forms. ) 

How  great  the  tendency  is  for  a  substance  to  remain  in  the  same 
form  after  passing  through  a  melting-temperature  or  transition -tem- 
perature and  thus  to  exist  in  an  unstable  form  depends  in  large 
measure  on  the  nature  of  the  substance.  The  following  general 
statements  in  regard  to  it  can,  however,  be  made.  A  crystalline 
solid  cannot  as  a  rule  be  heated  appreciably  above  its  melting- 
point;  thus,  ice  always  melts  sharply  at  0°  (under  a  pressure  of 
1  atm.).  On  the  other  hand,  a  liquid  (like  water)  can  ordinarily 
be  cooled  to  a  temperature  considerably  below  the  freezing-point  if 
agitation  and  intimate  contact  with  solid  particles,  especially  with 
the  stable  solid  phase,  is  avoided.  Still  more  pronounced  is  the 
tendency  of  solid  substances  to  remain  in  the  same  form  upon  being 
heated  or  cooled  through  a  transition-temperature;  thus  rhombic 
sulphur  can  be  heated  to  its  melting-point  (110°),  although  this  is 
about  15°  higher  than  the  transition-temperature  (95.5°)  at  which 
it  should  go  over  into  monoclinic  sulphur;  and  monoclinic  sulphur 
can  be  cooled  to  room  temperature  without  going  over  into  the 
rhombic  form,  provided  this  be  done  quickly  and  without  agi- 
tation. The  rate  at  which  an  unstable  phase  goes  over  into  the  stable 


98  PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 

one  tends  to  increase  with  the  distance  from,  the  transition-tempera- 
tures ;  but  when  the  substance  is  below  its  transition-temperature  this 
tendency  may  be  more  than  compensated  by  the  greatly  reduced  rate 
of  reaction  which  a  considerable  lowering  of  temperature  always  pro- 
duces; thus  white  phosphorus  is  an  unstable  form,  but  the  rate  at 
which  it  goes  over  into  the  stable  red  form  at  room  temperature 
is  so  small  that  it  may  be  preserved  unchanged  for  years;  similarly, 
diamond  is  an  unstable  form  of  carbon  at  room  temperature,  but  it 
does  not  go  over  into  graphite  or  amorphous  carbon. 

An  effective  means  of  causing  an  unstable  form  to  go  over  into 
a  stable  one  is  to  mix  it  intimately  with  the  stable  form.  The  transi- 
tion may  be  further  accelerated  by  moistening  the  mixture  of  the  two 
forms  with  a  solvent  in  which  they  are  somewhat  soluble.  These 
facts  are  made  use  of  in  the  determination  of  transition-tempera- 
tures. Thus  the  transition-temperature  of  sulphur  has  been  deter- 
mined by  charging  a  bulb  with  a  mixture  of  rhombic  and  monoclinic 
sulphur,  filling  it  with  carbon  bisulphide  and  oil  of  turpentine,  keep- 
ing it  for  an  hour  first  at  95°  and  then  at  96°,  and  noting  whether 
the  liquid  rose  or  fell  in  the  capillary  stem  attached  to  the  bulb. 
The  volume  was  found  to  decrease  steadily  at  95°  (owing  to  the 
transition  of  the  monoclinic  into  the  rhombic  form),  and  to  increase 
steadily  at  96°  (owing  to  the  reverse  transition),  showing  that  the 
transition-temperature  lies  between  95  and  96°. 

ProT).  9.  Suggest  an  explanation  of  the  catalytic  action  of  the  solvent 
in  accelerating  the  transition  of  the  sulphur. 

Pro  6.  10.  Outline  a  method  by  which  the  transition- temperature  of 
sulphur  could  be  determined  by  quantitative  solubility  measurements. 

THE  PHASE  RULE 

97.   Concept  of  Variance. 

Profc.  11.  If  sulphur  is  kept  at  a  specified  pressure  of  0.04  mm.,  at 
what  temperatures  is  it  stable,  a,  in  a  single  phase  as  rhombic  sul- 
phur, as  monoclinic  sulphur,  and  as  liquid  sulphur?  6,  in  two  phases, 
as  rhombic  and  monoclinic  sulphur,  and  as  monoclinic  and  liquid 
sulphur?  c,  in  the  three  phases,  rhombic  sulphur,  monoclinic  sulphur, 
and  sulphur- vapor? 

It  will  be  noted  that,  in  order  to  determine  the  position  on  the 
diagram  and  therefore  the  state  of  the  system,  the  values  of  two 


THE  PHASE  RULE  99 

determining  factors,  namely,  the  values  of  both  the  pressure  and 
the  temperature,  must  be  specified  when  there  is  only  one  phase; 
that  the  value  of  only  one  of  these  factors,  either  the  temperature 
or  pressure,  need  be  specified  when  any  two  phases  coexist;  and 
that  no  condition  can  be  arbitrarily  specified  when  any  three  phases 
coexist. 

The  number  of  determining  factors  whose  values  can  and  must 
be  specified  in  order  to  determine  the  state  of  a  system  consisting  of 
definite  phases  and  components  is  called  its  variance*;  and,  corre- 
sponding to  the  number  of  such  factors,  systems  are  said  to  be  non- 
variant,  univariant,  bivariant,  etc. 

It  is  evident  from  the  preceding  statements  that  when  a  one- 
component  system  consists  of  only  one  phase  the  system  is  bivariant, 
when  it  consists  of  two  phases  it  is  univariant,  and  when  it  consists 
of  three  phases  it  is  nonvariant.  In  other  words,  the  sum  of  the 
variance  and  number  of  phases  is  always  three  for  a  one-component 
system. 

Prob.  12.  Discuss  with  reference  to  the  principle  just  stated  and  to 
the  sulphur  diagram  the  possibility  of  the  coexistence,  a,  of  the  three 
phases,  rhombic,  monoclinic,  and  liquid  sulphur;  b,  of  the  four  phases, 
rhombic,  monoclinic,  liquid,  and  gaseous  sulphur. 


98.   Inductive  Derivation  of  the  Phase  Rule. 

Pro 6.  IS.  In  order  that  the  specific  properties  of  a  liquid  mixture  of 
ethyl  alcohol  and  water,  such  as  its  density,  specific  heat-capacity,  or 
refractive  index,  may  have  definite  values,  we  must  evidently  state  not 
only  that  it  is  at  some  definite  temperature  and  pressure  (say  20°  and 
1  atm.),  but  also  that  it  contains  some  definite  proportion  of  alcohol  or 
water  (say  30%  of  alcohol).  What  is  the  variance  of  a  system  consist- 
ing of  such  a  mixture?  What  are  the  number  of  phases  and  the  number 
of  components? 

Prob.  14.  In  the  case  of  each  of  the  following  systems,  state  of 
what  factors  the  values  might  be  specified  in  order  that  all  its  specific 
properties  may  be  fully  determined;  and  state  the  number  of  compo- 
nents, the  number  of  phases,  and  the  variance  of  the  system:  a.  Two 
solutions  produced  by  shaking  together  water  and  bromine.  6.  The 
two  solutions  of  water  and  bromine  and  their  vapors,  c.  The  two  solu- 
tions of  water  and  bromine,  their  vapors,  and  ice.  d.  A  solution  of  water, 

*Some  authors  use  the  expression  degrees  of  freedom,  Instead  of  the  term 
variance. 


100  PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 

ethyl  alcohol,  and  acetic  acid.  e.  A  solution  of  water,  ethyl  alcohol,  and 
acetic  acid  and  their  vapors. 

Prob.  15.  a.  Make  a  table  showing  the  number  of  components,  the 
number  of  phases,  and  the  variance  of  each  of  the  systems  named  in 
Probs.  11,  13,  and  14.  &.  State  the  effect  of  increasing  the  number  of 
phases  in  a  system  consisting  of  a  definite  number  of  components. 
c.  State  the  effect  of  increasing  the  number  of  components  in  a  system 
with  a  definite  number  of  phases,  d.  Add  to  the  table  a  column  showing 
the  sum  of  the  number  of  phases  and  of  the  variance  for  each  of  the 
systems. 

These  problems  show  that  in  every  case  the  sum  of  the  number  of 
phases  (P)  and  of  the  variance  (V)  is  greater  by  two  than  the  number 
of  the  components  (0) ;  that  is,  P  +  V  =  C  -f-  2.  This  principle,  which 
is  called  the  phase  rule,  is  a  general  one,  applicable  to  systems  consist- 
ing of  any  number  of  components  and  of  any  number  of  phases. 

The  phase  rule  furnishes  a  basis  for  the  classification  of  different 
types  of  equilibrium.  It  also  enables  the  number  of  phases  that  can 
exist  under  specified  conditions  to  be  predicted.  The  usefulness  of  the 
phase  rule  itself  is,  however,  often  exaggerated.  Of  primary  importance 
in  the  treatment  of  the  equilibrium  conditions  of  systems  in  relation  to 
the  phases  present  are  the  methods  of  representing  those  conditions  by 
diagrams,  as  described  in  later  articles  of  this  chapter. 

ProT).  16.  Sodium  carbonate  and  water  form  solid  hydrates  of  the 
composition  Na2CO3.H2O,  Na2CO3.7H2O,  and  Na2CO3.10H2O.  a.  How  many 
of  these  hydrates  could  exist  in  equilibrium  with  the  solution  and  ice 
under  a  pressure  of  1  atm.?  6.  How  many  of  these  hydrates  could  exist 
in  equilibrium  with  water- vapor  at  30°  ? 

99.  Derivation  of  the  Phase  Rule  from  the  Perpetual-Motion 
Principle.* — In  order  to  determine  fully  the  state  of  one-phase  systems 
consisting  of  any  number  C  of  components  (as  defined  in  Art.  94), 
evidently  the  composition  of  the  phase  and  in  addition  any  external 
factors  that  determine  its  properties  must  be  specified.  The  compo- 
sition of  the  phase  is  fully  determined  by  specifying  the  mol-fractions 
of  all  but  one  of  the  components,  that  is,  by  specifying  0  —  1  quantities. 
The  only  external  factors  which  commonly  affect  the  properties  of  a 
phase  of  specified  composition  are  temperature  and  pressure;  but 
in  special  cases  certain  other  factors,  some  of  which  are  mentioned 
below,  have  an  appreciable  influence-  Representing  the  number  of 

*  This  Article  may  be  omitted  in  briefer  courses. 


THE  PHASE  RULE  101 

such  external  determining  factors  by  n,  the  variance  or  total  num- 
ber of  quantities  (independent  variables)  that  must  be  specified  is 
C  —  1  -f-  n.  In  this  case,  therefore,  where  the  number  of  phases  P  is  one, 
the  sum  of  the  number  of  phases  and  the  variance  V  is  equal  to 
0  -j"  n>  that  is,  P  -f-  V  =  C  +  n;  or,  for  the  common  case  where 
pressure  and  temperature  are  the  only  external  determining  factors, 
P  -f-  V  =  C  +  2.  For  example,  the  state  of  one-phase  systems  con- 
sisting of  three  components  (1,  2,  3)  is  fully  determined  by  specifying 
four  quantities,  namely,  the  mol-fractions  (xv  and  x2)  of  any  two  of 
the  components  and  the  pressure  (p)  and  temperature  (T) ;  hence 
any  specific  property  whatever  of  the  system,  such  as  its  density  d, 
can  be  expressed  as  some  function  of  these  four  variables,  such  as 
d  =  f  (ajj,  xz,  p,  T).  In  the  general  case,  where  the  number  of  compo- 
nents is  C  and  the  number  of  external  factors  is  n,  the  function  becomes 
d  —  i(xlt  x2, . .  xc_v  p,  T, . .). 

The  derivation  of  the  phase-rule  now  consists  in  showing  that  the 
equation  P  -f-  V  =  C  +  n,  which  for  one-phase  systems  is  a  result 
of  the  definitions  of  components  and  of  external  factors,  still  holds 
true  whatever  be  the  number  of  phases.  In  order  that  this  may  be  so, 
it  is  evidently  necessary  only  that  each  new  phase  introduced  into  the 
system  shall  diminish  the  variance  by  one;  for  then  P  +  V  will  still 
have  its  former  value  C  +  n.  That  this  is  the  case  can  be  shown  as 
follows. 

Consider  that  a  system  of  C  components  (1,  2,  3,  . .  C)  exists  as  a 
gaseous  phase,  and  that  a  new,  liquid  or  solid,  phase  is  developed  in  it 
(for  example,  by  varying  the  pressure  or  temperature).  Now  the 
partial  pressures  (plf  p2,  . .  pc  )  of  the  separate  components  in  the 
gaseous  phase  are,  like  any  other  property  of  the  phase,  functions 
only  of  the  mol-fractions  (xlf  x2,  . .  xc_J  and  of  the  external  factors 
(p,  T,  . .).  This  conclusion  may  be  expressed  mathematically  as 
follows  :* 

P!  =  fi(#!,  #2>   •  •  XC  1>  Pt   T,  .  .)• 

p2  =  f,(xlt  x2,  . .  xc-i,  P,  T, . .). 


pc=  fc  (xv  x2,  . .  xc.l9  p,  T, . .). 

*  In  the  case  of  perfect  gases,  functions  of  the  simpler  form  Pi  =  a?i  p  hold 
true ;  but  the  other  variables  a?2,  •  •  T, . .  affect  pi  when  the  gases  do  not  conform 
to  the  perfect-gas  law. 


102         '  PtiASE  E'QUlLlB'RWM  IN  CHEMICAL  SYSTEMS 

When  the  new,  liquid  or  solid,  phase  is  present  in  equilibrium  with 
the  gaseous  phase,  the  partial  pressure  of  each  component  in  the 
gaseous  phase  is  determined  by  the  mol-fractions  (#/,  #/,  . .  x'c_j)  of 
the  liquid  or  solid  phase,  by  the  temperature,  and  by  any  other  external 
determining  factor  which  has  an  appreciable  influence ;  for,  if  a  liquid 
or  solid  phase  of  such  composition  as  to  be  in  equilibrium  with  the  gase- 
ous phase  could  also  be  in  equilibrium  at  the  same  temperature  with 
some  other  gaseous  phase  with  different  partial  pressures,  perpetual 
motion  of  the  kind  described  in  Art.  28  could  be  realized.  This  conclu- 
sion from  the  perpetual-motion  principle  that  the  partial  pressures 
must  be  fully  determined  by  the  mol-fractions  in  the  liquid  phase  and 
by  the  external  factors  may  be  expressed  mathematically  as  follows : 


PC  =    <*,      ,..  ac.lt  p,     ,... 

By  equating  these  two  sets  of  expressions  for  plt  p2,  ..pc,  the 
following  functional  relations  between  the  mol-fractions  in  the  gaseous 
phase  and  those  in  the  liquid  phase  are  obtained : 

i,(xl9  xv  . .  xc.l9  p,  T, . .)  =  f/«,  <, . .  *Vi,  P>  T>  •  -)- 
f2<X,  xv  .  -  arc.,,  p,  T, . .)  =  f2'<X',  <,  . .  *'c_i,  p,  T, . .). 

f  0  (xv  x2,  . .  xc.l9  p,  T,..}=  fc'  (a?/,  <, . .  xfc.,,  p,  T, . .). 

That  is,  the  new  phase  gives  us  C  new  functional  equations  in  which 
only  C — 1  new  variables  (namely,  x',  x2',  . .  #'c_i)  are  introduced. 
By  combining  these  new  equations  with  each  other  the  new  variables 
may  evidently  be  eliminated,  yielding  a  functional  relation  of  the  form : 

f(^,.T2,...Tc_rp,r,..)  =  0. 

This  is  obviously  a  relation  between  the  C  —  1  -f-  n  variables  which 
determine  any  property  of  the  gaseous  phase,  in  the  way  described  in 
the  first  paragraph  of  this  Article.  Hence  the  number  of  these  variables 
whose  values  can  and  must  be  specified  to  determine  the  properties  of 
the  gaseous  phase  is  one  less  than  it  was  when  that  phase  was  alone 
present.  And  in  a  similar  way  this  can  be  shown  to  be  true  also 
of  the  properties  of  the  liquid  or  solid  phase.  In  other  words,  the 
new  phase  diminishes  the  variance  of  the  system  by  one,  and  the  sum 


THE  PHASE  RULE  103 

P  -f-  V  retains  the  value  0  -f-  n  which  it  had  when  the  system  con- 
sisted of  a  single  phase.  And  evidently,  since  each  additional  phase 
formed  within  the  system  will  similarly  decrease  the  variance  by  one, 
the  sum  P  +  V  will  always  have  the  same  value  G  -\-  n. 

In  the  above  derivation  it  was  assumed  that  a  gaseous  phase  was 
present  in  the  system.  It  will  be  noted,  however,  that  the  partial 
pressures  in  the  gaseous  phase  were  employed  only  as  a  means  of 
deriving  functional  relations  between  the  mol-fractions  of  the  com- 
ponents in  two  different  phases,  and  that  the  partial  pressures 
disappeared  in  these  relations.  This  indicates,  and  it  can  be  rigorously 
shown,  that  functional  relations  between  the  mol-fractions  of  the 
same  form  as  those  given  above  hold  true  for  any  pair  of  phases ;  for 
example,  for  two  liquid  phases.  It  follows  therefore  that  the  phase 
rule  is  applicable  to  any  kind  of  system  whatever,  in  the  form 
P  -f  V  =  0  +  n. 

As  has  been  stated,  the  value  of  n  in  the  above  derived  expression 
of  the  phase  rule  is  commonly  2;  for  the  only  external  factors  that 
ordinarily  influence  appreciably  the  state  of  the  system  are  temper- 
ature and  pressure.  In  some  cases,  however,  other  factors  come  into 
play.  For  example,  this  is  sometimes  true  of  intensity  of  illumination 
or  of  electric  or  magnetic  field.  Thus  illumination  of  silver  chloride 
increases  its  dissociation-pressure;  and  an  electric  discharge  through 
an  equilibrium  mixture  of  nitric  oxide,  nitrogen,  and  oxygen  increases 
the  proportion  of  nitric  oxide  in  the  mixture.  Another  factor  which 
makes  the  value  of  n  greater  than  2  is  introduced  when  different 
pressures  are  applied  by  means  of  semipermeable  walls  to  different 
phases  of  the  system.  Thus  the  pressure  of  the  vapor  in  equilibrium 
with  a  liquid  is  progressively  increased  when  the  liquid  is  subjected 
to  an  increasing  pressure  by  means  of  a  piston  permeable  for  the 
vapor  only.  A  common  case  of  this  kind  is  that  where  the  atmosphere 
acts  as  such  a  piston,  exerting  a  pressure  on  the  liquid  and  solid  phases 
of  the  system,  but  not  on  the  components  in  the  vapor  phase. 


104  PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 

TWO-COMPONENT   SYSTEMS 

100.  Systems  with  Solid  and  Gaseous  Phases. — The  equilibrium  of 
systems  of  this  type  at  constant  temperature  has  already  been  considered 
from  the  mass-action  standpoint  in  Art.  91.  Their  equilibrium  in  rela- 
tion to  temperature  and  pressure  is  considered  from  the  phase-rule 
standpoint  in  the  following  problems. 

Prob.  11.  Silver  oxide  dissociates  fairly  readily  into  silver  and  oxy- 
gen, a.  Show  from  the  phase  rule  that  silver  oxide  can  be  heated  in 
oxygen  gas  at  a  given  pressure  through  a  certain  range  of  temperature 
without  any  decomposition  taking  place.  6.  Show  also  that  there  is  one 
temperature,  and  only  one,  at  which  a  mixture  of  silver  oxide  and  silver 
can  be  kept  under  oxygen  at  the  given  pressure  without  any  change 
taking  place. 

Prob.  18.  The  dissociation  pressure  of  silver  oxide  is  0.1  atm.  at  116°, 
0.2  atm.  at  132°,  1.0  atm.  at  175°,  and  2.0  atm.  at  197°.  a.  If  finely  divided 
silver  be  heated  in  the  air,  what  proportion  of  it  will  be  finally  converted 
into  oxide  when  the  temperature  is  130°  ?  when  it  is  140°  ?  6.  How  could 
silver  oxide  be  heated  to  170°  without  any  decomposition  taking  place? 

Prob.  19.  Describe  a  method  by  which  pure  oxygen  can  be  prepared 
from  the  air  with  the  aid  of  the  reaction  2BaO,  =  2BaO  -f  Oz. 

ProT).  20.  When  a  precipitate  of  hydrated  manganese  dioxide  is 
ignited  in  the  air  it  conies  to  a  constant  weight  corresponding  to  the  com- 
position MnO2  when  the  temperature  is  450°,  and  to  another  constant 
weight  corresponding  to  the  composition  Mn2O3  when  the  temperature  is 
500°  ;  but  when  ignited  in  oxygen  it  changes  to  MnO2  even  at  500°.  What 
do  these  facts  show  as  to  the  dissociation-pressures? 

Prob.  21.    The  dissociation-pressure  of  solid  calcium  carbonate 
is          10  ISO  320  580  760  1000    mm. 

at        600°  800°  840°  880°  896°  910° 

a.  At  what  temperature  will  it  begin  to  dissociate  when  it  is  heated  in  air 
free  from  carbon  dioxide?  &.  At  what  temperature  would  it  dissociate 
completely  when  heated  in  a  covered  crucible  (so  that  there  is  equaliza- 
tion of  the  pressure,  but  no  circulation  of  air  into  the  crucible)  ?  c.  In 
lime-burning  what  temperature  would  have  to  be  maintained  in  the  kiln 
if  there  were  no  circulation  of  gases  through  it?  What  temperature 
would  have  to  be  maintained  if  the  coal  used  as  fuel  were  burned  to 
carbon  dioxide  with  the  minimum  quantity  of  air  and  the  combustion- 
products  were  passed  up  through  the  kiln? 

Pro  b.  22.  A  method  has  been  suggested  for  the  standardization  of 
sulphuric  acid  solutions  which  consists  in  adding  an  excess  of  ammonia 
solution,  evaporating,  drying  the  residue  at  100°,  and  weighing.  In  dry- 
ing the  salt  some  decomposition  according  to  the  reaction  (NH4)2S04  = 
NH4HS04  -f-  A"//,  is  likely  to  take  place.  How  might  the  process  be  modi- 
fied so  as  to  hasten  the  drying  and  yet  entirely  prevent  the  decomposition ': 


TWO-COMPONENT  SYSTEMS  105 

101.  Systems  with  Solid,  Liquid,  and  Gaseous  Phases.  Pressure- 
Temperature  Diagrams. — When  a  two-component  system  consists  of 
three  phases,  the  phase  rule  evidently  shows  that  the  specifica- 
tion of  one  of  the  determining  factors  (for  example,  the  temperature) 
fixes  the  state  of  the  system  and  therefore  the  values  of  the  other  factors 
(for  example,  of  the  pressure  and  of  the  composition  of  the  liquid  or 
the  gaseous  phase).  The  pressures  at  which  the  different  groups  of 
three  phases  exist  in  equilibrium  at  various  temperatures  can  therefore 
be  represented  by  lines  on  a  diagram  in  which  the  pressure  and 
temperature  are  taken  as  the  coordinates. 

Figure  8  shows  a  portion  of  the  pressure-temperature  diagram  for 
a  system  consisting  of  the  two  components  disodium  hydrogen  phos- 
phate (Na2HPO4)  and  water  (H2O),  for  the  case  that  the  vapor-phase 
(V),  which  consists  only  of  water-vapor,  is  always  present.  These  two 
components  are  known  to  form  the  following  solid  phases:  ice  (I), 
anhydrous  salt  (A),  dihydrate  Na2HPO4.2H2O  (AW2),  heptahydrate 
Na2HP04.7H3O  (AWT),  and  dodecahydrate  Na2HPO4.12H,O  (AWJ ; 
also  a  solution-phase  (S)  of  variable  composition,  approaching  pure 
water  as  one  limit. 

Prob.  23.  Show  from  the  phase  rule  how  many  phases,  in  addition  to 
the  vapor-phase,  must  be  present  in  a  two-component  system  in  order 
that  the  pressure  of  the  vapor  may  have  a  definite  value  at  any  given 
temperature. 

Pro 6.  24.  At  30°  one  formula-weight  of  Na2HPO4  is  placed  in  contact 
with  a  large  volume  of  water-vapor  at  1  mm.,  and  the  volume  of  the 
vapor  is  steadily  diminished  (so  slowly  that  equilibrium  is  established) 
until  finally  there  remains  in  contact  with  the  vapor  only  the  saturated 
solution  (which  contains  the  components  in  the  proportion  !Na2HPO4  : 
33.2  H2O ) .  State,  with  the  aid  of  Figure  8,  the  changes  that  take  place 
in  the  pressure  of  the  vapor  and  the  accompanying  changes  that  take 
place  in  the  character  of  the  other  phases  in  contact  with  it,  throughout 
the  whole  process. 

Prob.  25.  Plot  the  pressure  of  the  vapor  (as  ordinates)  against  the 
number  of  formula-weights  of  water  absorbed  by  the  salt  as  abscissas 
for  the  process  described  in  Prob.  24  (up  to  the  point  where  15  formula- 
weights  have  been  absorbed).  Mark  the  lines  on  the  plot  so  as  to  show 
what  phases  are  present  during  each  stage  of  the  process. 

Pro b.  26.  Make  a  plot  like  that  of  Prob.  25  for  the  case  that  the 
process  described  in  Prob.  24  takes  place  at  38°  (instead  of  at  30°). 

Prob.  27.  Describe  a  method  of  determining  what  hydrates  of  copper 
sulphate  exist  at  25°. 


106 


PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 


Prob.  28.  Describe  with  the  aid  of  Figure  8  what  changes  take  place 
when  a  mixture  of  heptahydrate  and  dodecahydrate  is  heated  from  30° 
to  38°  in  a  sealed  tube  previously  evacuated. 

Prob.  29. — Conditions  under  which  Salts  are  Efflorescent  or  Hygro- 
scopic.—At  30°  moist  air  is  in  contact  with  solid  Na2HPO4.7H2O.  Under 
what  conditions  of  humidity,  »,  would  the  salt  remain  unchanged? 
o,  would  it  lose  water?  c,  would  it  absorb  water?  (By  the  humidity  of  a 
gas  is  meant  the  ratio  of  the  pressure  of  the  water- vapor  in  the  gas  to 
the  vapor-pressure  of  water  at  the  same  temperature.) 


45 


40 


35  - 


15 


10 


5- 


A-V 


TEMPERATURE 


25° 


40° 


FIGURE  8 


Pro6.  30. — Preparation  of  Pure  Hydrates. — a.  State  the  conditions 
under  which  moist  crystals  of  Na2HPO4.12H2O  could  be  completely  dried 
at  30°  without  any  danger  of  decomposition.  6.  Describe  a  method  by 
which  these  conditions  could  be  practically  realized. 


TWO-COMPONENT  SYSTEMS  107 

Separation  of  Hydrates  from  Solutions. — 

Prob.  31.  State  what  solid  phase  separates  on  evaporating  a  dilute 
solution  of  Na2HPO4,  a,  at  30°,  b,  at  38°. 

Prob.  32.  The  equilibrium-pressures  (p)  for  the  reaction  CaS04.2H20 
(gypsum)  =  CaS04  (anhydrite)  -f-  2H2O  and  the  vapor-pressures  (p0)  of 
pure  water  at  various  temperatures  (t)  are  as  follows: 

t  50°  55°  60°  65° 

p  80  109  149  204  mm. 

Po  92  118  149  188  mm. 

The  solubility  of  calcium  sulphate  is  so  small  that  the  vapor-pressure  of 
its  saturated  solution  may  be  considered  to  be  identical  with  that  of  water. 
a.  State  what  happens  on  heating  gypsum  from  50°  to  65°  in  a  sealed 
tube  previously  evacuated,  b.  State  what  solid  phase  separates  when 
a  solution  of  calcium  sulphate  is  evaporated  at  55°,  and  at  65°.  c.  State 
what  solid  phase  would  separate  upon  evaporating  the  solution  at  55°  if, 
when  it  became  saturated,  enough  calcium  chloride  were  added  to  reduce 
its  vapor-pressure  by  10%.  Give  the  reasons  in  each  case. 

102.  Systems  with  Solid  and  Liquid  Phases.  Temperature-Com- 
position Diagrams. — The  conditions  under  which  three  phases  of  a  two- 
component  system  coexist  may  be  also  represented  by  a  diagram  in 
which  the  coordinates  are  the  temperature  and  the  composition  of  any 
phase  in  which  both  components  are  present  in  measurable  quantities. 
Such  diagrams  are  often  employed  when  one  of  the  phases  is  liquid. 

From  the  phase-rule  standpoint  it  is  evident  that  the  equilibrium 
conditions  of  a  two-component  system  can  still  be  represented  by  a 
temperature-composition  diagram  when  the  specification  that  the 
vapor-phase  is  present  is  replaced  by  the  specification  that  the  pressure 
has  some  definite  value  (greater  than  that  at  which  the  vapor  can 
exist).  Moreover,  since  pressure  has,  as  illustrated  in  Art.  95,  only  a 
small  effect  on  equilibria  in  which  solid  and  liquid  phases  are  alone 
involved,  the  lines  on  the  temperature-composition  diagram  have  sub- 
stantially the  same  position  when  the  system  is  under  a  pressure  of  one 
atmosphere  as  they  do  when  it  is  under  the  pressure  of  the  vapor. 
And  in  practice  composition-temperature  diagrams  are  ordinarily  con- 
structed from  data  determined  under  the  atmospheric  pressure. 

The  form  of  the  temperature-composition  diagram  varies  greatly 
with  the  character  of  the  solid  phases  which  the  components  are  capable 
of  producing.  The  simplest  type  of  such  a  diagram  is  that  in  which  the 
two  components  A  and  B  do  not  form  any  solid  compound  with  each 
other,  but  separate  from  the  solution  in  the  pure  state.  This  type  is 


108 


PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 


illustrated  by  Figure  9,  which  shows  the  complete  diagram  for  the 
systems  composed  of  acetic  acid  (A)  and  benzene  (B). 

The  freezing-point  curve  CD  represents  the  composition  of  the 
solutions  (S)  which  are  in  equilibrium  with  solid  A,  and  the  freezing- 
point  curve  DF  represents  the  composition  of  the  solutions  which  are 
in  equilibrium  with  solid  B  at  different  temperatures.  The  point  D  at 
which  the  solution  is  in  equilibrium  with  the  two  solid  phases  A  and  B 
is  called  the  eutectic  point.  When  a  solution  in  the  condition  corre- 
sponding to  this  point  is  cooled,  it  solidifies  completely  without  change 


-10° 


-15° 


FIGURE 


of  composition  or  temperature  to  a  mixture  of  the  two  solid  phases  A 
and  B.  This  mixture  is  usually  so  fine-grained  and  intimate  that  it 
differs  markedly  in  texture  from  ordinary  mixtures  of  the  same  solid 
phases.  It  is  called  the  eutectic  mixture,  or  simply  the  eutectic. 

Pro&.  33.  Show  that  the  phase  rule  requires  that  a  solution  at  the 
eutectic  point  solidify  without  change  of  temperature  or  composition 
when  heat  is  withdrawn  from  it. 

Pro6.  84.  A  tube  containing  a  solution  of  80%  benzene  and  20% 
acetic  acid  at  10°  is  placed  within  an  air  jacket  surrounded  by  a  freezing 
mixture  at  — 20°,  so  that  the  system  slowly  loses  heat,  till  its  temperature 
falls  to  — 10°.  Predict  with  the  aid  of  Figure  9  the  values  of  the  tem- 
perature and  composition  of  the  solution  at  which  any  phase  appears  or 
disappears.  State  also  the  character  of  the  solid  mixture  finally  obtained. 


TWO-COMPONENT  SYSTEMS  109 

Prob.  35. — Cooling  Curves. — a.  On  a  diagram  having  as  ordinates  the 
temperatures  in  degrees  and  as  abscissas  the  time  of  cooling  in  arbitrary 
units  draw  curves  representing  in  a  general  way  the  rate  at  which  the 
temperature  decreases  when  a  solution  of  80%  benzene  and  20%  acetic 
acid  is  cooled  as  described  in  Prob.  34,  assuming,  first,  that  the  liquid 
overcools  without  the  separation  of  any  solid  phase;  and  assuming, 
secondly,  that  the  solid  phases  separate  so  that  there  is  always  equilib- 
rium. (Take  into  account  the  fact  that  on  cooling  a  system  there  is 
always  an  evolution  of  heat  whenever  a  new  phase  separates.)  6.  Draw 
on  the  same  diagram,  at  the  right  of  these  curves,  a  new  cooling  curve 
showing  how  the  temperature  changes  when  pure  benzene  is  cooled  from 
10°  to  — 10°.  c.  Draw  a  cooling  curve  also  for  the  case  that  a  solution 
of  04%  benzene  and  36%  acetic  acid  is  cooled  from  10°  to  — 10°. 

The  fields  in  the  diagram  are  also  of  much  significance  since  the 
composition  represented  by  the  abscissas  is  understood  to  be  that  of  the 
whole  system — not  merely  that  of  the  liquid*  phase.  Thus,  when  the 
system  consists  of  the  substances  in  such  a  proportion  (e.  g.,  20%  of  B 
to  80%  of  A)  and  at  such  a  temperature  (e.  g.,  0°)  that  its  condition 
is  represented  by  a  point  n  within  the  field  CDG,  the  diagram  shows 
that  it  consists  of  the  two  phases,  solid  A  and  solution  S  of  the  compo- 
sition (corresponding  to  the  point  p)  at  which  these  two  phases  are  in 
equilibrium  at  the  given  temperature.  Similarly,  when  the  system  has 
such  a  composition  and  temperature  that  it  lies  within  the  field  GD  JH, 
it  consists  of  solid  A  and  the  eutectic  mixture  E^,  which  always  has 

o 

the  composition  corresponding  to  the  point  D.  It  can  readily  be  shown, 
moreover,  that,  when  the  state  of  the  system  is  represented  by  the 
point  n,  the  weight  of  solid  A  is  to  the  weight  of  the  solution  present 
as  the  length  of  the  line  joining  n  and  p  is  to  the  length  of  the  line 
joining  m  and  n ;  and  similarly,  that  at  any  point  below  r  on  the  same 
ordinate  with  it  the  weight  of  pure  A  is  to  the  weight  of  the  eutectic 
mixture  present  as  the  length  of  the  line  rD  is  to  that  of  the  line  Gr. 

Prol).  36.  How  do  the  two  rectangular  fields  at  the  bottom  of  Figure  9 
differ  with  respect  to,  a,  the  phases  present ;  6,  the  texture  of  the  mixture? 

Prob. -SI.  Show  that  the  relation  stated  in  the  last  sentence  of  the 
preceding  text  is  true. 

Prob.  38.  Lead  (which  melts  at  327°)  and  silver  (which  melts  at 
960°)  form  a  eutectic  which  melts  at  304°.  The  heat  absorbed  by  the 
fusion  of  one  atomic  weight  of  lead  is  1340  cal.  Calculate  the  composi- 
tion of  the  eutectic,  taking  into  account  the  facts  that  the  first  part  of  a 
freezing-point  curve  can  be  located  with  the  aid  of  the  laws  of  perfect 
solutions,  and  that  the  molecules  of  metallic  elements  in  dilute  metallic 


110 


PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 


solutions  are  as  a  rule  identical  with  their  atoms.  Compare  the  calculated 
composition  of  the  eutectic  with  that  (4.7  at.  wts.  Ag  to  95.3  at.  wts.  Pb) 
derived  from  cooling  curves. 

Pro  ft.  89.  Describe  a  method  based  on  the  facts  stated  in  Prob.  38 
by  which  a  melt  containing  1  at.  wt.  Ag  and  99  at.  wts.  Pb  could  be  en- 
riched in  silver.  State  the  extent  to  which  the  percentage  of  silver  could 
be  increased  by  the  method  described.  (A  method  of  this  kind  has,  In 
fact,  been  employed  by  metallurgists.) 

Pro&.  40-  Construction  of  Temperature-Composition  Diagrams  from 
Cooling  Curves. — Figure  10  shows  the  cooling  curves  for  a  series  of  mix- 
tures of  magnesium  and  lead  containing  the  atomic  percentages  of  lead 
shown  by  the  numbers  at  the  tops  of  the  curves.  On  a  diagram  whose 
coordinates  are  temperature  and  atomic  percentages  plot  points  repre- 
senting the  temperature  at  which  the  solidification  of  each  mixture  begins 
and  ends.  Draw  in  solid  lines  representing  the  freezing-point  curves. 
Draw  also  dotted  lines  limiting  the  different  fields  (as  was  done  in 
Figure  9),  and  letter  the^elds  so  as  to  show  of  what  the  system  consists 
in  each  field. 


700C- 


600°- 
550°  • 
500°- 

450° 
400° 
350° 
300° 
250°  • 


700° 
650° 
600° 
550° 
500° 
450° 
400° 
350° 
300° 
250° 


FIGURE  10 

Pro  6.  41.  Construct  a  temperature-composition  diagram  for  the  sys- 
tem composed  of  Na2HPO4  and  H2O  by  plotting  the  following  values  of 
the  percentage  (lOOa?)  of  Na2HPO4  in  the  saturated  solution  as  abscissas 
against  the  temperature  (t)  as  ordinates.  The  solid  phase  which  is  in 


TWO-COMPONENT  SYSTEMS  111 

equilibrium  with  the  solution  (S)  at  a  pressure  of  1  atm.  is  indicated 
by  the  letters  above  the  data. 

Ice                  AWia  AW12                  AW12  AW12             AW, 

100#          1.1                 2.4  5.5                  19.2  30.0            33.7 

t         —0.5°               0°  15°                 30°  35°              38° 

AW-,  AW7  AW 2  AW2  AW2 

100#        38.0  43.5  44.7  40.6  48.0 

t  43°  48°  50°  55°  60° 

Draw  in  on  the  diagram  lines  representing  the  equilibrium  conditions, 
and  mark  each  line  with  letters  indicating  the  phases  which  coexist 
under  the  conditions  represented  by  it. 

Pro 6.  42.  With  the  aid  of  the  diagram  of  Prob.  40  determine  all  the 
conditions  of  temperature  and  composition  at  which  three  phases  coexist, 
stating  in  each  case  what  the  three  phases  are. 

Prob.  43.  State  with  the  aid  of  the  diagram  of  Prob.  40  what  changes 
occur  in  the  composition  of  the  solution,  and  what  solid  phases  separate 
and  redissolve,  on  cooling  from  55°  to  — 1°,  at  1  atm.,  a  solution  contain- 
ing, a,  1%  Na2HP04;  ft,  20%  Na2HPO4;  c,  46%  Na2HPO4. 

Prob.  44.  What  solid  hydrate  separates  when  a  solution  containing 
15%  Na2HPO4  is  evaporated  (for  example,  by  passing  a  current  of  air 
through  it)  at  30°?  at  40°?  at  50°? 

Pro 6.  45.  With  the  aid  of  the  diagram  of  Prob.  40,  estimate  the  ratio 
of  the  solubility  of  Na2HPO4.7H2O  to  that  of  Na2HPO4.12H2O  at  30°, 
expressing  the  solubilities  in  formula-weights  Na2HPO4  per  1000  grams 
of  water. 

Pro6.  ^6.  Prove  that  in  contact  with  the  solution  the  more  soluble 
hydrate,  Na2HPO4.7H2O,  is  unstable  at  30°  with  respect  to  the  less  soluble 
one,  Na2HPO4.12H2O,  by  showing  what  must  happen  a,  when  the  hepta- 
hydrate  is  placed  in  contact  with  a  solution  saturated  at  30°  with  the 
dodecahydrate ;  &,  when  the  dodecahydrate  is  placed  in  contact  with  a 
solution  saturated  at  30°  with  the  heptahydrate. 

Proft.  47-  0"  Predict  from  the  diagram  at  what  temperature  the 
hydrate  Na2HPO4.12H2O  would  melt  if  the  transition  into  Na2HPO4.7H2O 
were  avoided.  6.  Insert  on  the  diagram  dotted  lines  and  letters  showing 
(as  in  Figure  9)  the  significance  of  the  different  fields. 

Profe.  48. — Correlation  of  the  Pressure-Temperature  and  Tempera- 
ture-Composition Diagrams. — With  the  aid  of  the  diagrams  of  Figure  8 
and  of  Prob.  40  make  a  table  showing  for  the  temperatures  20,  25,  30, 
35.5,  and  40°  the  vapor-pressures  and  percentage  compositions  of  the 
saturated  solutions  and  the  nature  of  the  solid  phases  with  respect  to 
which  the  solutions  are  saturated.  (Note  that  the  effect  of  pressure 
on  the  solubility  is  here  neglected. ) 

Profc.  49.— Systems  with  Solid  Phases  and  Two  Liquid  Phases. — 
Tn  Figure  11  the  curves  CD  and  DE  show  the  atomic  percentages  of 
the  two  liquid  phases  in  equilibrium  with  each  other  at  various  temper- 


112 


PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 


atures.  a.  State  all  that  would  happen  on  cooling  from  1000°  to  300° 
alloys  containing  40  and  90  atomic  percents  of  zinc.  6.  State  all  that 
would  happen  on  gradually  adding  zinc  to  molten  lead  at  350°,  at  700°, 
and  at  1000°. 

Note. — Two  liquid  phases  are  often  formed  also  by  pairs  of  non- 
metallic  substances,  such  as  bromine  and  water,  or  phenol  and  water. 


1000°-, 

-1000° 

-JS. 

900°- 

^''*       X\ 

-900° 

••*                 S                \ 

/'                  \ 

800°- 

-800° 

'                        \ 

/                         \ 

700°- 

\ 

-700° 

/                              \ 

/                                I 

600°  - 

/ 

-600° 

/ 

500°- 

/ 

-500° 

c/                              E 

400°- 

^B/' 

-400° 

300°  - 

F  |                                         G 
H  !  J                                       K 

-300° 

1     i     i     i     i     i     i     i     i 

Pb  K 

K)    90    80    70    60    50    40    30    20    10 

D 

Zn   ( 

)     10    20    30    40    50    60    70    80    90    1 

DO 

FIGURE  11 


103.  Systems  with  Liquid  and  Gaseous  Phases. — Systems  of  two 
components  having  a  gaseous  phase  and  a  liquid  phase  in  which  the 
components  are  miscible  in  all  proportions  have  already  been  con- 
sidered in  Arts.  32  and  33.     They  are  briefly  reviewed  in  Prob.  50 
with  respect  to  their  diagrammatic  representation.    The  case  where 
the  two  components  have  only  limited  miscibility  in  the  liquid  state 
was  considered  in  Probs.  74  and  75,  Art.  39. 

Prob.  50.  Sketch  three  separate  diagrams  corresponding  to  the  three 
cases  represented  in  the  temperature-composition  diagram  in  Figure  3, 
Art.  33,  where  the  pressure  on  the  system  was  fixed  at  1  atm.  On  these 
sketches  letter  each  of  the  fields  so  as  to  show  of  what  phases  the  system 
consists  when  represented  by  any  point  on  the  diagram. 

104.  Systems  involving  Solid  Solutions. — The  components  some- 
times separate  from  the  liquid  solution  in  the  form  of  solid  solutions, 


TWO-COMPONENT  SYSTEMS  113 

instead  of  in  the  form  of  the  pure  solid  substances  or  of  solid  com- 
pounds of  them.  By  solid  solutions  are  meant  physically  homogeneous 
solid  mixtures  of  two  or  more  substances,  that  is,  solid  mixtures  which 
contain  no  larger  aggregates  than  the  molecules  of  the  substances.  In 
their  equilibrium  relations  they  closely  resemble  liquid  solutions; 
differing  from  them  mainly  in  the  respect  that  the  equilibrium  condi- 
tions are  less  readily  established,  owing  to  the  rigidity  and  inertness 
characteristic  of  the  solid  state.  Each  component  of  a  solid  solution 
lowers  the  vapor-pressure  of  the  other  component  according  to  the 
same  principles  as  in  the  case  of  liquid  solutions. 

Pro  6.  51.  The  addition  of  thiophene  to  benzene  raises  the  freezing- 
point  of  benzene  (instead  of  lowering  it),  a.  Show  how  the  fact  that 
these  two  substances  form  solid  solutions  with  each  other  might  account 
for  the  raising  of  the  freezing-point,  by  considering  the  vapor-pressure 
relations  as  in  Art.  34.  6.  In  which  solution  must  the  mol-fraction  of 
the  thiophene  be  greater  in  order  that  the  freezing-point  may  be  raised  ? 

The  two  components  are  sometimes  soluble  in  each  other  in  the  solid 
state  in  all  proportions,  forming  a  complete  series  of  solid  solutions. 
In  other  cases  each  component  has  only  a  limited  solubility  in  the 
other  solid;  so  that  two  series  of  solid  solutions  result,  each  covering 
only  a  limited  range  of  composition. 

Proft.  52. — Alloys  with  Complete  Series  of  Solid  Solutions. — The 
cooling  curves  show  that  on  cooling  molten  mixtures  of  nickel  and  copper 
solidification  begins  and  becomes  complete  at  the  following  temperatures, 
a  solid  solution  separating  in  each  case : 

Percentage  of  nickel  0  10  40  70  100 

Solidification  begins  1083°        1140°        1270°         1375°        1452° 

Solidification  ends  1083°        1100°        1185°        1310°        1452° 

a.  Draw  on  a  composition-temperature  diagram  two  continuous  curves 
corresponding  to  these  data.  Letter  each  field  so  as  to  show  of  what  the 
system  consists  at  any  point  within  it.  6.  State  what  happens  on  slowly 
cooling  a  50%  mixture  from  1400°  to  1200°,  giving  the  compositions  of 
the  liquid  and  solid  solutions  in  equilibrium  with  each  other  at  the  tem- 
peratures at  which  solidification  begins  and  ends,  and  at  1275°. 

Note. — It  will  be  seen  that  the  melting-point-composition  curves  of 
the  preceding  problem  are  entirely  similar  to  the  boiling-point-compo- 
sition curves  marked  I  in  Figure  3,  Art.  33.  Melting-point-composition 
curves  are  also  met  with  analogous  to  the  curves  marked  II  and  III 
in  that  figure,  in  which  the  broken  lines  now  represent  the  composition 
of  the  liquid  phase  and  the  continuous  lines  the  composition  of  the  solid 
solution  in  equilibrium  with  it. 


114 


PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 


Pro&.  53.  Gold  and  copper  form  a  complete  series  of  solid  solutions. 
One  of  the  mixtures  containing  60  atomic  percent  gold  has  a  constant 
melting-point  of  880°.  Gold  melts  at  1063°,  and  copper  at  1083°. 
a.  Sketch  the  temperature-composition  diagram,  as  in  Prob.  52a.  &.  What 
indication  does  the  diagram  afford  of  the  existence  of  a  compound  of 
gold  and  copper? 

Pro&.  54. — Alloys  with  Limited  Series  of  Solid  Solutions. — In 
Figure  12,  the  diagram  for  silver-magnesium  alloys,  in  which  the  com- 
position is  expressed  in  atomic  percentages,  the  curves  KD,  DL,  and  MF 


1000°- 


900°- 


_1000° 


-900° 


_800° 


-700° 


-600° 


-500° 


400° 


FIGURE  12 


represent  the  composition  of  solid  solutions  in  equilibrium  with  the 
liquid  solutions  whose  composition  is  represented  by  the  curves  CD,  DE, 
and  EF,  respectively,  a.  State  what  equilibria  are  represented  by  each 
of  the  other  curves  on  the  diagram  and  by  the  lines  GH,  CK,  and  LM. 
6.  State  what  compounds  are  indicated  by  the  diagram,  c — h.  State 
what  happens  on  cooling  slowly  till  complete  solidification  results  a 
liquid  mixture  containing  the  following  atomic  percentages  of  silver: 
c,  90 ;  d ,  70 ;  e,  55 ;  /,  50 ;  g,  25 ;  h,  20.  i.  Specify  the  phases  in  which 
the  system  exists  when  its  composition  and  temperature  are  represented 
by  any  point  in  each  of  the  fields  CKD,  DLE,  EMF,  QKDLR,  RLMT, 
and  BCJH. 


THREE-COMPONENT  SYSTEMS  115 

THREE-COMPONENT   SYSTEMS 

105.  Systems  with  Gaseous,  Liquid,  and  Solid  Phases.  Temper- 
ature-Composition Diagrams. 

Applications  of  the  Phase  Rule. — 

Pro  6.  55.  a.  Discuss  with  reference  to  the  phase  rule  the  effect  of 
temperature  and  of  pressure  on  the  solubility  of  calcium  carbonate 
in  water  saturated  with  carbon  dioxide  gas.  6.  Derive  from  the  equi- 
librium laws  applicable  to  dilute  solutions  a  quantitative  expression 
showing  how  the  solubility  of  calcium  carbonate  varies  with  the  pressure 
of  the  carbon  dioxide  gas. 

Prob.  56*.  Silver  chloride  forms  with  ammonia  two  solid  compounds 
AgCLl^NHa  and  AgC1.3NH8.  a.  Show  by  the  phase  rule  under  what 
conditions  AgCl  exists  in  contact  with  an  aqueous  solution  and  its  vapor 
at  25°  ;  6,  under  what  conditions  AgCl  and  AgCl.l^NH3  so  exist ;  c,  under 
what  conditions  AgCl,  AgCl.liNH3,  and  AgC1.3NH3  so  exist,  d.  At  25° 
ammonia  vapor  is  in  equilibrium  with,  the  solids  AgCl  and  AgCl.l£NH3 
at  10  mm.  and  with  the  solids  AgCl.l$NH3  and  AgC1.3NH3  at  105  mm. 
Show  quantitatively  from  these  data  under  what  conditions  each  of 
these  two  pairs  of  compounds  would  exist  in  equilibrium  with  an 
aqueous  solution,  e.  Referring  to  the  data  of  Prob.  35,  Art.  27,  what 
can  be  said  as  to  the  concentration  of  the  aqueous  solution  at  which 
AgCl  is  converted  into  AgCl.l£NH3.  /.  Assuming  that  the  AgCl  in 
the  solution  is  substantially  all  in  the  form  of  completely  ionized 
Ag(NH3)2+Cl-,  derive  from  the  equilibrium  laws  of  dilute  solutions  a 
relation  between  the  solubility  s  of  silver  chloride  and  the  partial 
pressure  p  of  NH3  in  the  vapor,  first,  when  the  solid  phase  is  AgCl,  and 
secondly,  when  it  is  AgCl.l^NH3. 

It  will  be  seen  from  the  preceding  problems  that  the  phase  rule 
shows  that  the  solubility  at  any  definite  temperature  is  some  function 
of  the  pressure,  and  that  the  equilibrium  laws  of  dilute  solution  (the 
distribution-law  and  mass-action  law)  show  what  that  function  is, 
provided  the  molecular  forms  in  which  the  components  exist  in  the 
solution  are  known.  This  is  a  characteristic  difference :  the  phase  rule 
is  qualitative,  and  its  application  presupposes  no  special  knowledge 
beyond  that  of  the  number  of  components  and  phases,  and  it  is  appli- 
cable without  any  limitation.  The  mass-action  law  is  quantitative, 
its  application  presupposes  knowledge  of  the  molecular  species  present, 
and,  if  numerical  values  are  to  be  computed,  also  of  their  dissociation- 
constants;  and  it  is  applicable  in  exact  form  only  to  solutions  or 
gases  at  small  concentrations. 


116 


PHASE  EQUILIBRIUM  IN  CHEMICAL  SYSTEMS 


The  most  common  method  of  plotting  the  composition  of  three- 
component  systems,  which  has  the  advantage  of  treating  the  three 
components  symmetrically,  is  to  make  use  of  a  diagram  consisting 
of  an  equilateral  triangle,  along  the  three  sides  of  which  are  plotted 
the  percentages  or  mol-fractions  of  the  three  components,  as  illus- 
trated in  Figure  13.  In  such  a  diagram  the  vertices  then  represent 


AAAXV 


/OO 


100 


60         SO         40 

Per  Cent  Sn 

FIGURE  13 

the  pure  components,  points  on  the  sides  represent  mixtures  of  each 
pair  of  components,  and  points  within  the  triangle  represent  mixtures 
of  all  three  components.  Thus  the  upper  vertex  would  represent  pure 
cadmium;  the  point  H,  a  mixture  consisting  of  42 (atomic) %  Cd  and 
58%  Sn;  and  the  point  G,  a  mixture  consisting  of  22%  Cd,  57%  Sn, 
and  21%  Pb. 

At  any  one  temperature  the  various  compositions  of  the  liquid 
phase  with  which  any  definite  solid  phase  is  in  equilibrium  are  repre- 
sented on  such  a  diagram  by  a  line.  Thus  in  Figure  13  the  lines 
HJ,  JK,  and  LM  represent  at  190°  the  compositions  of  the  liquid 


THREE-COMPONENT  SYSTEMS  117 

with  which  solid  cadmium,  solid  tin,  and  solid  lead,  respectively,  are 
in  equilibrium.  The  point  J  then  evidently  represents  the  compo- 
sition of  the  liquid  with  which  the  two  solid  phases  Cd  and  Pb  are 
in  equilibrium  at  190°.  Similar  lines  can  be  drawn  corresponding  to 
various  other  temperatures,  thus  giving  a  series  of  isotherms  repre- 
senting the  effect  of  temperature  on  the  equilibrium  of  the  phases. 
The  dotted  line  FG  evidently  shows  the  variation  with  the  temperature 
of  the  composition  of  the  liquid  phase  in  equilibrium  with  the  two 
solid  phases  Cd  and  Pb;  the  line  DG  shows  the  same  thing  for  the 
two  solid  phases  Cd  and  Sn;  and  the  line  EG  for  the  two  solid  phases 
Sn  and  Pb.  And  the  point  G  shows  the  only  temperature  and  liquid 
composition  at  which  the  three  solid  phases  Cd,  Pb,  and  Sn  coexist 
in  equilibrium  with  the  liquid  phase.  The  mixture  of  three  solid  phases 
separating  at  this  point  is  called  the  ternary  eutectic,  and  the  point 
itself  the  ternary  eutectic  point. 

A  more  complete  representation  of  the  effect  of  temperature  is 
secured  by  plotting  temperatures  along  an  axis  perpendicular  to  the 
plane  of  the  composition  triangle.  A  prismatic  model  thus  results, 
whose  horizontal  sections  are  the  isotherms  represented  in  a  plane 
triangular  diagram,  like  that  in  Figure  13. 

Profe.  57.  A  completely  liquid  mixture  of  50 (atomic)  %  Cd,  30%  Sn, 
and  20%  Pb  is  cooled  till  it  wholly  solidifies,  a.  State  the  temperature 
at  which  solidification  begins  and  the  nature  of  the  solid  phase  which 
then  separates.  &.  State  the  direction  on  the  diagram  which  on  further 
cooling  the  changing  composition  of  the  liquid  phase  follows.  (Note 
that  the  separation  of  the  solid  phase  does  not  change  the  ratio  of  the 
atomic  quantities  of  the  other  two  components  in  the  liquid  phase.) 
c.  State  the  temperature  at  which  a  second  solid  phase  begins  to  sepa- 
rate, the  composition  of  the  liquid  phase,  and  the  nature  of  the  solid 
phase,  a.  Describe  what  happens  on  further  cooling,  e.  Describe  the 
texture  of  the  solid  alloy. 


CHAPTER   VIII 

THERMOCHEMISTRY:  THE  PRODUCTION  OF  HEAT  BY 
CHEMICAL  CHANGES 


THE   FUNDAMENTAL  PRINCIPLES    OP  THERMOCHEMISTRY 

106.  Introduction. — This  chapter  is  devoted  to  a  consideration  of 
the  heat-effects  that  attend  changes  in  the  physical  state  or  chemical 
composition  of  substances.    The  branch  of  chemistry  treating  of  these 
heat-effects  is  called  thermochemistry.     The  chapter  is  divided  into 
two  main  parts.    The  first  part,  entitled  the  fundamental  principles 
of  thermochemistry,  is  devoted  to  the  energy  principles  underlying 
the  subject  and  to  the  general  methods  of  determining  and  expressing 
thermochemical  quantities.    The  second  part,  entitled  the  results  of 
thermochemistry,   is   devoted  to   the  generalizations   that   have   been 
derived  from  the  results  of  thermochemical  measurements  in  the  case 
of  substances  in  the  different  physical  states. 

107.  The  Law  of  the  Conservation  of  Energy,  or  the  First  Law  of 
Energetics. —  The  essential  idea  involved   in  the  concept  of  energy 
is  the  constancy  of  a  quantity  which  is  involved  in  all  the  changes 
taking  place  in  the  universe;  and  this  is  often  explicitly  expressed  by 
the  statement  that  energy  is  neither  created  nor  destroyed  in  any 
process  whatever.    This  statement  is  called  the  Law  of  the  Conservation 
of  Energy,  or  the  First  Law  of  Energetics. 

The  law  may  be  stated  more  concretely  as  follows :  When  a  quantity 
of  energy  disappears  at  any  place,  a  precisely  equal  quantity  of  energy 
simultaneously  appears  at  some  other  place  or  places;  and  when  a 
quantity  of  energy  disappears-  in  any  form,  a  precisely  equal  quantity 
of  energy  simultaneously  appears  in  some  other  form  or  forms;  equal 
quantities  of  energy  of  different  forms  .being  understood  to  be  such 
quantities  as  produce  the  same  effect  (for  example,  in  modifying 
motion  or  raising  temperature)  when  converted  into  the  same  form. 

The  exactness  of  this  law  has  been  established  by  many  careful 
quantitative  investigations  made  for  the  purpose.  The  law  is  also 

118 


FUNDAMENTAL  PRINCIPLES  119 

confirmed  by  the  correspondence  of  the  conclusions  drawn  from  it  with 
well-established  facts  and  principles.  Among  these  may  be  mentioned 
as  especially  important  the  following  principle,  which  is  a  conclusion 
based  upon  the  failure  of  many  attempts  to  produce  a  contrary  result : 
The  production  of  an  unlimited  amount  of  work  by  a  machine  or 
arrangement  of  matter  which  receives  no  energy  from  the  surround- 
ings is  an  impossibility.  An  ideal  process  like  that  here  stated  to  be 
impossible  is  sometimes  called  perpetual  motion  of  the  first  kind  (to 
distinguish  it  from  another  kind  of  perpetual  motion  which  will  be 
described  in  the  next  chapter). 

108.  Energy-Changes  in  the  Surroundings  attending  Changes  in  the 
State  of  a  System. — The  composition  of  matter  under  consideration  is 
termed  the  system.  A  system  is  said  to  be  in  a  definite  state  when  the 
temperature,  pressure,  state  of  aggregation,  quantity,  and  chemical 
composition  of  each  of  its  parts  is  fixed;  and  a  change  in  any  of  these 
conditions  is  called  a  change  in  state. 

It  follows  from  the  First  Law  that,  when  the  state  of  a  system  is 
fixed,  its  energy-content  U  is  also  fixed;  and  therefore  also  that  any 
change  in  the  state  of  a  system  is  attended  by  a  definite  change  A  £7 
in  its  energy-content  (equal  to  the  difference  U2  —  U^  between  its 
energy-content  in  the  two  states),  whatever  be  the  process  by  which 
the  change  takes  place.  This  corollary  from  the  First  Law  of  Ener- 
getics, stating  that  the  change  in  the  energy-content  of  a  system  is 
determined  solely  by  its  initial  and  final  states,  is  of  so  much  impor- 
tance in  thermochemical  considerations  that  it  has  received  a  special 
name — the  law  of  initial  and  final  states. 

Corresponding  to  the  change  in  the  energy-content  of  the  system, 
there  must,  of  course,  be  an  energy  effect  in  the  surroundings.  Tbe 
energy  lost  or  gained  by  the  system  may  appear  or  disappear  in  the 
surroundings  in  any  of  its  various  forms;  but  in  energetic  considera- 
tions it  is  primarily  important  only  to  differentiate  the  production 
of  heat  from  that  of  the  other  forms  of  energy.  Such  other  forms  of 
energy  (that  may  be  associated  with  matter)  are  collectively  desig- 
nated work.  Under  this  term  are  included,  for  example,  production  of 
motion  in  a  body,  displacement  of  a  force  through  a  distance,  change 
of  volume  under  pressure,  development  of  electrical  energy,  and 
production  of  chemical  changes.  All  these  forms  of  work  are  quanti- 


120  THERMOCHEMISTRY 

tatively  transformable  into  one  another;  but  the  transformation  of 
heat  into  work  is  subject  to  certain  limitations,  which  will  be  considered 
later.  It  is  for  this  reason  that  heat  and  work  are  differentiated  from 
each  other.  It  will  be  noted  that  the  term  work  is  here  used  in  a  sense 
different  from  that  in  which  it  is  used  in  the  science  of  mechanics. 

The  various  units  in  which  energy-quantities  are  expressed  and  the 
relations  between  them  were  denned  in  Art.  22. 

Pro6.  1.  When  12  g.  of  carbon  are  burned  at  20°  within  a  closed 
vessel  (so  that  no  work  is  produced)  with  oxygen  forming  carbon  dioxide, 
97,000  cal.  are  evolved ;  and  when  28  g.  of  carbon  monoxide  are  so 
burned  with  oxygen,  67,700  cal.  are  evolved.  Show  by  the  law  of  initia-l 
and  final  states  how  the  heat  evolved  by  the  burning  of  12  g.  of  carbon 
to  carbon  monoxide  can  be  calculated. 

Prob.  2.  When  1  g.  of  liquid  water  is  vaporized  at  100°  and  1  atm., 
the  heat  withdrawn  from  the  surroundings  is  537  cal.,  and  the  work 
produced  by  the  expansion  against  the  constant  pressure  of  1  atm.  is 
168  joules.  At  100°  and  1  atm.  which  has  the  greater  energy-content, 
1  g.  of  liquid  water  or  1  g.  of  water-vapor,  and  what  is  the  difference 
between  the  two  energy-contents  in  calories? 

With  the  aid  of  the  concepts  of  energy-content,  heat,  and  work, 
the  First  Law  may  now  be  expressed  by  the  statement  that  the  increase 
Af7  in  the  energy-content  of  the  system  is  equal  to  the  heat  Q  with- 
drawn from  the  surroundings  diminished  by  the  work  W  produced  in 
the  surroundings ;  that  is, 

U2  —  U,  =  AC7  ==  Q  --  W. 

It  is  to  be  noted  that  Q  always  represents  the  heat  withdrawn  from 
the  surroundings,  and  W  the  work  produced  in  them;  and  that  there- 
fore, when  actually  heat  is  produced  in  the  surroundings  or  work  is 
withdrawn  from  them,  the  numerical  value  of  Q  or  W  is  negative. 

Thermochemistry  deals  with  the  heat-effects  Q  and  the  changes  in 
energy-content  ACT"  which  attend  changes  in  the  state  of  systems. 
Of  these  two  quantities  the  change  in  energy-content  is  more  funda- 
mental, since  it  has  a  definite  value  for  any  definite  change  in  state; 
while  the  heat-effect  varies  with  the  quantity  of  work  which  may  be 
produced,  and  therefore  with  the  process  by  which  the  change  in  state 
takes  place,  for  example,  whether  it  takes  place  in  a  closed  vessel,  or 
under  the  pressure  of  the  atmosphere,  or  in  a  voltaic  cell.  In  thermo- 


FUNDAMENTAL  PRINCIPLES  121 

chemical  considerations,  however,  the  only  form  of  work  commonly 
involved  is  that  corresponding  to  a  change  in  volume  of  the  system 
under  a  constant  pressure.  This  is  considered  in  Art.  109. 

Proft.  3.  a.  What  is  the  system  considered  in  Prob.  2?  6.  What  is 
the  change  in  state?  (In  describing  the  "change  in  state"  the  initial 
and  final  states  should  always  be  explicitly  stated.)  c.  What  is  the  value 
of  AC/  for  this  change  in  state?  d.  What  is  the  process  by  which  the 
change  in  state  takes  place?  e.  What  are  the  values  of  Q  and  of  W  for 
this  process?  /.  By  what  other  process  (involving  the  production  of  no 
work)  could  the  same  change  in  state  be  brought  about?  g.  What  would 
be  the  values  of  AC/  and  of  Q  in  this  case? 

109.  Work  attending  Changes  in  Volume. — The  work  produced 
when  a  system  changes  its  volume  is  most  readily  derived  for  the  case 
that  the  volume  undergoes  a  change  in  dimensions  in  one  direction 
only.  Suppose  that  a  liquid  or  gaseous  substance  contained  in  a 
cylinder  is  enclosed  by  a  movable  piston  of  cross-section  a,  and  that 
a  force  /  is  exerted  upon  this  piston,  for  example,  by  a  weight  placed 
upon  it,  just  sufficient  to  compensate  the  expansive  force  of  the  body 
and  prevent  its  expansion.  Suppose  now  that  the  external  force  be 
reduced  by  an  infinitesimal  amount  and  that  the  piston  rises  through 
a  distance  dl.  The  increase  of  volume  dv  is  then  a  dl,  and  the  expansive 
force  acting  upon  the  unit  of  surface,  which  force  is  called  pressure  p, 
is  f/a.  The  work  dW  produced  by  the  expansion  is  therefore  given 
by  the  equations : 

dW  =  f  dl  =  p  a  dl  =  p  dv. 

That  is,  the  work  is  equal  to  the  product  of  the  pressure  into  the 
infinitesimal  increase  of  volume  that  takes  place.  It  can  be  easily 
demonstrated  that  this  equation  also  holds  true  in  the  general  case  in 
which  the  volume  increases  in  dimensions  in  any  number  of  directions. 
The  general  expression  for  the  work  produced  when  a  body  under- 
goes a  change  of  volume  from  v1  to  vv  is  therefore: 

rv* 

W=    (      P  dv. 


It  is  evidently  necessary  to  know  the  functional  relation  between 
pressure  and  volume  before  the  integration  can  be  carried  out. 


122  THERMOCHEMISTRY 

When  the  pressure  is  constant  during  the  change  of  volume,  the 
equation  evidently  becomes : 

W  =  p  (v2  -  Vl). 

Pro&.  4'  Calculate  the  work  in  ergs  and  in  calories  produced  by  the 
vaporization  of  one  formula- weight  of  water  at  100°  against  the  pressure 
of  1  atin.  The  specific  volume  of  liquid  water  at  100°  is  1.043,  and  that 
of  saturated  water-vapor  at  100°  is  1650. 

Pro 6.  5.  a.  Formulate  an  exact  expression  (in  terms  of  the  volumes 
involved)  for  the  work  produced  when  A7  mols  of  hydrogen  are  produced 
by  dissolving  an  equivalent  quantity  of  zinc  in  dilute  sulphuric  acid  solu- 
tion at  2'°  at  a  constant  pressure  p.  &.  Simplify  this  expression  by 
neglecting  the  relatively  small  volumes  involved  and  by  applying  the 
perfect-gas  law.  c.  Calculate  the  work  in  ergs  and  calories  produced  when 
1.008  grams  of  hydrogen  are  produced  at  20°. 

Pro&.  6.  Calculate  the  work  in  calories  produced  when  the  reaction 
CO  -f  iO2  =  CO2  takes  place  at  20°  and  1  atm. 

Pro  6.  7.  a.  Show  that,  in  general,  when  any  change  in  state  which 
takes  place  at  constant  temperature  and  pressure  is  attended  by  an 
increase  £N  in  the  number  of  mols  of  the  gaseous  substances  present, 
the  work  produced  in  the  surroundings  is  given  approximately  by  the 
expression  W  =  A  2V  R  T.  ?>.  Give  the  value  of  A  A7  in  this  expression 
for  the  change  of  state  involved  in  each  of  the  three  preceding  problems. 

Pro 6.  8.  Calculate  the  work  in  calories  produced  when  1  mol  of 
oxygen  at  pressure  p  and  temperature  T  is  heated  at  constant  pressure 
through  one  degree. 

110.  Heat-Effects  attending  Changes  in  Temperature. — A  compara- 
tively simple  kind  of  change  of  state  is  that  which  a  system  undergoes 
when  its  temperature  is  increased.  The  ratio  of  the  quantity  of  heat 
dQ  absorbed  when  its  temperature  rises  from  T  to  T  -\-  dT  to  the  rise 
of  temperature  dT  is  called  its  heat-capacity  ((7)  at  T° ;  that  is,  C  = 
dQ/ 'dT.  The  heat-capacity  is  substantially  equal  to  the  quantity  of 
heat  absorbed  when  the  temperature  rises  one  degree.  When  the  heat- 
ing takes  place  without  change  of  pressure,  the  ratio  dQ/dT  is  called 
the  heat-capacity  at  constant  pressure  Cp.  When  the  heating  takes 
place  without  change  of  volume,  the  ratio  is  called  the  heat-capacity 
at  constant  volume  Cv.  The  heat-capacity  of  any  system  is  the  sum  of 
the  heat-capacities  of  its  homogeneous  parts;  and  the  heat-capacity 
of  any  such  part  is  the  product  of  its  weight  by  the  heat-capacity  of 
one  gram  of  it,  which  is  called  its  specific  heat-capacity  C.  The  heat- 
capacity  of  one  atomic  weight,  one  mol,  or  one  formula-weight  of  a 
pure  substance  is  called  its  atomic,  molal,  or  formal  heat-capacity. 


FUNDAMENTAL  PRINCIPLES  123 

Profc.  9.  When  100  g.  of  silver  at  100°  are  immersed  in  1000  g.  of 
water  at  15.000°  the  temperature  rises  to  15.475°.  Calculate  the  heat- 
capacity  of  this  weight  of  silver  and  the  specific  and  atomic  heat-capacities 
of  silver,  assuming  these  quantities  to  be  constant  between  15°  and  100°. 

Prob.  10.  The  molal  heat-capacity  of  oxygen  at  constant  pressure  at 
the  temperature  T  is  given  by  the  expression  M  "Cp  =  6.50  -f  0.0010T. 
Calculate  the  heat  absorbed  in  heating  2.24  liters  of  oxygen  at  0°  and 
1  atm.  to  100°,  the  pressure  remaining  constant. 

111.  Heat-Effects  attending  Changes  in  State  at  Constant  Temper- 
ature.—  The  heat  withdrawn  from  or  imparted  to  the  surroundings 
when  a  change  takes  place  in  the  state  of  a  system  which  is  kept  at  a 
constant  temperature  is  experimentally  determined  by  calorimetric 
measurements,  which  involve  the  principles  illustrated  by  the  follow- 
ing problem. 

Pro 6.  11.  Into  a  calorimeter  containing  50H2O  at  20.00°  1KC1  at 
20.00°  is  introduced,  and  the  temperature  falls  to  15.11°.  a.  What  change 
in  state  takes  place  in  this  process,  considering  the  calorimeter  to  be  a 
part  of  the  system ;  and  what  is  the  change  in  the  energy-content  of  the 
system,  neglecting  the  small  quantity  of  work  produced  and  any  loss  of 
heat  by  radiation?  b.  What  change  in  state  takes  place  when  1KC1  is 
dissolved  in  50H2O  at  20°  ?  c.  In  order  to  calculate  the  heat-effect  attend- 
ing this  change  in  state,  what  other  change  in  state  must  be  combined 
with  that  occurring  in  the  calorimeter?  d.  State  what  additional  data 
would  be  needed  to  calculate  this  heat-effect,  and  formulate  an  expres- 
sion by  which  the  calculation  could  be  made.  e.  Calculate  the  value  of 
this  heat-effect  with  the  aid  of  such  of  the  following  data  as  may  be 
needed:  the  heat-capacity  of  the  calorimeter  is  19  cal.  per  degree;  the 
specific  heat-capacity  of  solid  potassium  chloride  is  0.166,  of  water  is 
1.00,  and  that  of  the  solution  of  1KC1  in  50H2O  is  0.904  cal.  per  degree. 

Changes  in  state  at  constant  temperature  may  take  place  either 
without  change  of  volume  or  without  change  of  pressure.  The  different 
heat-effects  attending  these  two  different  changes  in  state  are  known 
as  the  heat-effect  at  constant  volume  Q  and  the  heat-effect  at  constant 
pressure  Q  p. 

The  heat-effect  at  constant  pressure  is  the  one  which  is  usually 
called  the  heat  of  reaciionf  and  the  one  which  is  commonly  recorded 
in  tables  of  constants.  There  is  difference  of  usage  regarding  the 
algebraic  sign  of  the  heat  of  reaction:  when  heat  is  actually  evolved 
by  the  reaction,  the  heat  of  reaction  is  usually  taken  positive  in  thermo- 
chemical  considerations,  but  negative  in  thermodynamic  considera- 


124  THERMOCHEMISTRY 

tions.  In  this  book  a  uniform  convention,  corresponding  to  the  thermo- 
dynamic  one,  is  employed  throughout;  heat-effects  being  always  con- 
sidered positive  when  heat  is  absorbed  (as  in  the  vaporization  of  water) 
and  negative  when  it  is  evolved  (as  in  the  combustion  of  hydrogen  and 
oxygen). 

112.  Changes  in  Energy- Content  and  in  Heat-Content  attending 
Changes  in  State. — Having  discussed  the  determination  of  the  work 
and  heat  produced  in  the  surroundings  by  processes  involving  a  change 
in  the  state  of  a  system,  the  corresponding  change  taking  place  in  the 
energy-content  of  the  system  itself  may  be  now  considered.  When 
the  change  in  state  takes  place  at  constant  volume,  it  is  evident  that 
no  work  is  involved,  and  that  therefore  the  heat  absorbed  0  from  the 

V 

surroundings  is  equal  to  the  increase  U2  —  U1  in  the  energy-content 
of  the  system.  When,  on  the  other  hand,  the  change  takes  place  at 
constant  pressure,  there  is  not  only  a  quantity  of  heat  Qp  withdrawn 
from  the  surroundings,  but  also  a  quantity  of  work  equal  to  p  (v2  —  vj 
produced  in  them.  The  increase  U2  —  U^  in  the  energy-content  of 
the  system  is  then  equal  to  Qp  —  p  (v2  —  vj.  The  change  in  energy- 
content  may  thus  be  determined  for  changes  in  state  which  can  be 
made  to  take  place  either  at  constant  volume  or  at  constant  pressure. 
Instead  of  determining  and  recording  the  values  of  U2  —  Ul  for 
various  changes  of  state,  it  is  generally  more  convenient  to  employ 
the  values  of  the  quantity  (U2  +  p2  v2)  —  (Ut  +  p^  vj.  The  quan- 
tity U  +  p  v,  like  the  quantity  U,  is  a  property  of  the  system  which 
always  has  a  definite  value  when  the  system  is  in  a  definite  state, 
and  which  always  changes  in  value  by  a  definite  amount  when  the 
system  changes  from  one  state  to  another,  whatever  be  the  process  by 
which  the  change  in  state  is  brought  about ;  for  it  is  evident  that  the 
pressure  p  and  the  volume  v,  as  well  as  the  energy-content  U,  have 
values  which  are  determined  by  the  state  of  the  system.  In  other 
words,  a  law  of  initial  and  final  states  applies  to  the  change  in  the 
quantity  U  +  p  v,  just  as  it  does  to  the  change  in  the  quantity  U. 
For  brevity,  this  quantity  U  -\-  p  v  will  be  represented  by  a  single 
letter  H,  and  will  be  called  the  heat-content  of  the  system,  it  being 
understood  that  this  term  is  a  purely  conventional  one  which  does  not 
imply  that  the  energy  quantity  denoted  by  it  is  a  heat  quantity,  any 
more  than  the  term  energy-content  implies  it. 


FUNDAMENTAL  PRINCIPLES  125 

For  any  change  in*  state  taking  place  by  any  process  whatever,  the 
change  in  heat-content  may  be  found  by  subtracting  from  the  heat- 
effect  attending  the  process  the  work  produced  by  it  diminished  by  the 
increase  in  the  pressure-volume  product;  that  is,  in  general: 


=  Q  —  [W  —  AO)]. 

For  a  change  in  state  for  which  the  initial  and  final  pressures  p, 
and  p2  have  the  same  value  p,  the  increase  AH  in  the  heat-content  is 
equal  to  the  heat-effect  Qp  when  the  change  takes  place  at  the  constant 
pressure  p;  that  is,  it  is  equal  to  the  heat  of  reaction  commonly 
employed. 

Pro&.  12.  Derive  the  two  conclusions  stated  in  the  last  two  para- 
graphs of  the  preceding  text. 

Pro&.  13.  a.  When  the  reaction  2  CO  -f-  O2  =  2  CO2  takes  place 
without  change  of  temperature  or  pressure  in  a  system  consisting  of 
2  mols  of  CO  and  1  mol  of  O,  at  20°  and  1  atm.,  the  heat-effect  is 
—  136,000  cal.  What  is  the  increase  in  the  heat-content,  and  what  is  the 
increase  in  the  energy-content,  of  the  system?  6.  When  the  reaction  takes 
place  in  the  same  mixture  without  change  of  temperature  or  volume,  the 
heat-effect  is  —  135,420  cal.  What  is  the  increase  in  the  heat-content  and 
what  is  the  increase  in  the  energy-content  in  this  case?  c.  In  what  re- 
spect does  the  final  state  of  the  system  in  a  differ  from  that  in  6,  and 
what  conclusion  can  be  drawn  from  a  comparison  of  the  results  obtained 
in  a  and  &  as  to  the  change  in  energy-content  and  in  heat-content  that 
would  attend  the  change  from  the  final  state  in  a  to  the  final  state  in  &? 
(It  will  be  seen  later  that  such  a  conclusion  is  justifiable  only  when  the 
system  consists  of  a  perfect  gas.  ) 

113.  Expression  of  Heat-Effects  by  Thermochemical  Equations.  — 
In  order  to  express  the  changes  in  heat-content  that  attend  changes  in 
state,  especially  those  involving  chemical  reactions,  at  any  constant 
temperature  and  pressure,  equations  are  conveniently  employed  in 
which  the  heat-contents  of  the  various  substances  involved  are  repre- 
sented by  their  chemical  formulas,  and  in  which  the  change  in  heat- 
content  is  shown  by  placing  a  numerical  term  on  the  right-hand  side 
of  the  equation.  For  example,  the  expression. 

Fe203  +  3  CO  =  2  Fe  -f  3  C02  +  9000  cal.  (at  20°) 

signifies  that  at  20°  and  1  atm.  (this  pressure  being  understood  unless 
some  other  pressure  is  stated)  the  heat-content  of  one  formula-weight 
of  ferric  oxide  plus  that  of  three  formula-weights  of  carbon  monoxide 


126  THERMOCHEMISTRY 

is  equal  to  the  heat-content  of  two  formula-weights  of  iron  plus  that  of 
three  formula-weights  of  carbon  dioxide  plus  9000  cal. ;  9000  cal.  being 
the  decrease  ( —  A5T)  in  the  heat-content  of  the  system,  which  is 
equivalent  to  the  heat  evolved  by  the  system  when  the  reaction  takes 
place  at  a  constant  temperature  and  pressure.  Such  expressions  are 
called  thermochemical  equations,  or  specifically,  heat-content  equations. 
As  indicated  in  the  preceding  equation,  the  fact  that  a  substance 
is  in  the  solid  state  is  shown  by  black-face  type,  and  the  fact  that  it  is 
in  the  gaseous  state  by  italics.  The  fact  that  a  substance  is  liquid 
is  denoted  by  ordinary  type,  and  the  fact  that  a  substance  is  dissolved 
in  x  formula-weights  of  water  is  shown  by  attaching  the  symbol  #Aq 
to  the  formula  of  the  substance.  Thus  the  equation 

KOI  +  100 Aq  ==  KCllOOAq  —  4400  cal.  (at  20°) 

signifies  that  when  at  20°  one  formula- weight  of  solid  potassium 
chloride  is  dissolved  in  100  formula-weights  of  water  there  is  an  in- 
crease of  4400  cal.  in  the  heat-content  of  the  system,  corresponding 
to  an  absorption  of  4400  cal.  from  the  surroundings.  When  the  sub- 
stance is  dissolved  in  so  large  a  quantity  of  water  that  the  addition 
of  more  water  produces  no  appreciable  heat-effect,  the  symbol  c°  Aq  may 
be  attached  to  the  formula  of  the  substance. 

These  thermochemical  equations  can  evidently  be  treated  strictly 
as  algebraic  equations,  and  can  be  combined  with  one  another  by  addi- 
tion or  subtraction;  for  every  quantity  in  them  has  a  definite  value 
(namely,  that  of  the  heat-content  of  the  substance  represented  by  the 
formula),  irrespective  of  the  other  quantities  that  occur  with  it  in 
the  equations. 

Profc.  14.  The  union  at  20°  and  1  atm.  of  1  g.  of  aluminum  with 
oxygen  is  attended  by  a  heat-evolution  of  7010  cal. ;  and  the  union  of  1  g. 
of  carbon  with  oxygen  to  form  carbon  monoxide  is  attended  by  a  heat- 
evolution  of  2420  cal.  Express  these  data  in  the  form  of  thermochemical 
equations ;  and  calculate  from  them  the  heat  of  the  reaction  A1208  -{-  3  C 
=  2  Al  -f  3  CO  at  20°. 

Prob.  15.  a.  Express  the  following  data  in  the  form  of  thermo- 
chemical equations,  employing  the  conventions  described  in  the  preceding 
text :  The  heat  of  formation  of  1  mol  gaseous  HC1  from  the  elementary 
substances  is  — 22,000  cal.  Its  heat  of  solution  in  100  formula-weights 
of  water  is  — 17,200.  The  heat  of  the  reaction  between  1  mol  gaseous 
chlorine  and  a  solution  of  2  formula-weights  HI  in  200  formula-weights 


FUNDAMENTAL  PRINCIPLES  127 

of  water,  forming  solid  iodine  and  a  dilute  HC1  solution  is  — 52,400  cal. 
The  heat  of  solution  of  1  mol  gaseous  HI  in  100  formula-weights  of  water 
is  — 19,200  cal.  &.  By  combining  these  equations  calculate  the  heat  of 
formation  of  1  mol  gaseous  HI  from  gaseous  hydrogen  and  solid  iodine. 

Although  it  is  true  that  only  changes  in  heat-content  can  be  deter- 
mined, yet  it  is  convenient  to  employ  an  arbitrary  scale  of  heat-content 
which  has  as  its  zero-points  the  heat-contents  of  the  various  elementary 
substances  at  the  temperature  under  consideration,  at  a  pressure  of  one 
atmosphere,  and  in  the  form  which  is  most  stable  at  this  temperature 
and  pressure.  Under  this  convention  the  heat-content  (£T)  of  any  com- 
pound substance  is  evidently  equal  to  the  increase  in  heat-content  (AH) 
which  attends  its  formation  out  of  the  elementary  substances;  and  in 
any  thermochemical  equation  the  formula  of  a  substance  may  evidently 
be  replaced  by  the  numerical  value  of  its  heat  of  formation.  For 
example,  the  heat-content  at  20°  of  one  formula- weight  of  gaseous 
hydrogen  bromide,  or  the  numerical  value  of  the  formula  ~\.HBr,  is 
— 8500  cal. ;  for  — 8500  cal.  is  the  heat  absorbed  when  it  is  formed  out 
of  gaseous  hydrogen  and  liquid  bromine  at  20°  and  1  atmosphere. 
Similarly  the  heat-content  of  a  potassium  chloride  solution  represented 
by  the  formula  KCllOOAq  is  —101,200  cal.;  for  this  is  the  sum  of 
the  heat-effect  ( — 105,600  cal.)  attending  the  formation  of  one  formula- 
weight  of  solid  KC1  out  of  solid  potassium  and  gaseous  chlorine  at  20° 
and  of  the  heat-effect  (4,400  cal.)  attending  its  solution  in  25  formula- 
weights  of  water  at  20°.  This  example  of  the  heat  of  formation  of  a 
solution,  which  may  be  expressed  by  the  equation  K  -\-  %C12  +  100 Aq 
=  KCllOOAq  +  101,200  cal.,  shows  that  in  thermochemical  equations 
the  symbol  #Aq  (not  attached  to  another  formula)  has,  like  the  formu- 
las of  elementary  substances,  the  value  zero;  for  the  water  represented 
by  it  has  not  been  formed  out  of  its  elements.  (Water  which  has  been 
so  formed  is  represented  by  the  formula  #H2O). 

It  is  evident  that  the  employment  of  heats  of  formation  greatly 
simplifies  the  task  of  determining  and  systematizing  thermochemical 
data;  for,  instead  of  measuring  and  recording  the  change  in  heat- 
content  attending  every  chemical  reaction,  it  suffices  to  do  this  for  the 
formation  of  every  compound  out  of  the  corresponding  elementary 
substances.  The  numerical  values  of  the  heats  of  formation  so  deter- 
mined may  then  be  substituted  in  any  thermochemical  equation,  and 


128  THERMOCHEMISTRY 

the  change  in  heat-content  attending  the  reaction  expressed  by  it  may 
thus  be  calculated. 

Prol).  16.  Calculate  the  heat-effect  that  attends  at  20°  the  reaction 
PbS  +  2PbO  —  3Pb  4-  &O2,  from  the  following  heats  of  formation  at 
20° :  PbO,  —50,300  cal. ;  PbS,  —19,300  cal. ;  8Oy  —70,200  cal. 

Prob.  17.  At  20°  the  heat  of  combustion  of  one  mol  of  acetylene 
(C,H,)  is  — 313,000  cal.  Calculate  its  heat  of  formation.  The  heat  of 
formation  of  1H2O  is  — 68,400  cal.,  and  that  of  1CO2  from  charcoal  and 
oxygen  is  — 96,600  cal. 

Prol).  18.  a.  State  just  what  heats  of  formation  are  denoted  by  the 
second  and  third  terms  in  the  following  equation : 

Zn  -|-  2HClooAq  =  ZnCl,coAq  4.  H3  -f-  34,200  cal.  (at  20°). 
6.  Write  the  complete  thermochemical  equations  which  express  these 
heats  of  formation,  taking  into  account  the  facts  that  the  heat  of  forma- 
tion of  1  mol  of  gaseous  HC1  is  — 22,000  cal.  and  that  its  heat  of  solution 
in  a  large  quantity  of  water  is  — 17,300  cal. 

114.  Indirect  Determination  of  the  Heat-Effects  of  Chemical 
Changes. —  On  account  of  radiation-errors  the  heat-effect  can  be 
directly  determined  by  calorimetric  measurements  only  for  those 
chemical  changes  which  take  place  completely  within  a  few  minutes, 
and  for  such  changes  only  when  the  temperature  is  not  greatly  different 
from  the  room-temperature.  It  is,  however,  possible  to  calculate  the 
heat-effects  of  many  other  changes  from  those  which  have  been  directly 
measured,  by  applying  the  law  of  initial  and  final  states.  This  is  illus- 
trated for  changes  at  the  room-temperature  by  the  problems  of  the 
preceding  articles  and  by  the  following  problems,  which  are  solved  by 
combining  the  proper  thermochemical  equations  in  such  a  way  as  to 
eliminate  the  heat-content  of  all  the  substances  except  those  involved 
in  the  reaction  under  consideration.  The  method  commonly  employed 
for  determining  heat-effects  at  temperatures  much  higher  or  lower  than 
the  room-temperature  is  described  in  the  next  article. 

Prob.  19.  Calculate  the  heat  of  formation  of  one  formula-weight  of 
solid  zinc  hydroxide  from  the  following  equation  and  from  the  other 
necessary  data,  which  have  been  given  in  preceding  problems : 

Zn(OH),  4-  2HC1  °°  Aq  =  ZnC^ooAq  4.  2H2O  -f   19,900  cal. 

Prob.  20.  Calculate  the  heat  of  formation  at  20°  of  one  formula- 
weight  of  H2SO4  from  the  following  data  at  20°  and  those  given  in  pre- 
ceding problems:  the  heat  of  solution  in  a  large  quantity  of  water  of 
1  mol  gaseous  SO,  is  —8000  cal.,  and  that  of  1  formula-weight  H2SO, 
is  — 18,000  cal.  One  mol  gaseous  Cl,  acting  OD  a  dilute  solution  of  1  mol 


FUNDAMENTAL  PRINCIPLES  129 

SO,  with  formation  of  a  dilute  solution  of  HC1  and  HaSO4  produces  a 
heat-evolution  of  73,900  cal.  (Note  that  when  x  formula- weights  of  H2O 
are  involved  in  the  chemical  reaction  the  symbol  a?H2O  must  appear  In 
the  thermochemical  equation,  even  though  o>Aq  may  also  occur  in  it.) 

Profc.  21.  A  direct  determination  of  the  heat-effect  of  the  reaction 
CH8CH,OH  oo  Aq  -{-  O2  =  CH8CO2H  co  Aq  -f  H2O  is  not  practicable.  State 
what  measurements  could  be  made  which  would  enable  this  heat-effect 
to  be  calculated ;  and  show  how  it  would'  be  calculated  from  the  results  of 
such  measurements. 

Pro 6.  22.  Suggest  a  series  of  chemical  reactions  from  whose  heat- 
effects,  which  must  be  readily  determinable  in  a  calorimeter,  the  heat 
of  formation  of  Na2CO3.10H2O  at  20°  could  be  calculated.  Write  the 
thermochemical  equations,  and  indicate  how  they  would  be  combined  to 
yield  the  desired  result. 

Prob.  23.  Calculate  the  heat-effects  of  the  following  reactions  at 
room-temperature,  referring  to  Landolt-Bornstein  Tabellen  for  the  heats 
of  formation,  of  combustion,  and  of  solution  needed : 

a.    H28  -f  2Ag  =  Ag2S  -f  H¥ 

1).    CgH^Oc   (glucose)   =  2C2H6OH  -f  20O2. 

c.    Zn  -f-  CuSO4400Aq  =  Cu  -f-  ZnSO4400Aq. 

115.  Influence  of  Temperature  on  the  Heat-Effects  attending 
Chemical  Changes. — The  heat-effect  at  constant  pressure  of  a  chemical 
change  taking  place  at  any  temperature  can  be  derived  from  the  heat- 
effect  at  constant  pressure  at  any  other  temperature  by  the  following 
consideration  of  two  different  processes  resulting  in  the  same  change 
in  state.  In  one  process  cause  the  chemical  change  (for  example,  the 
union  of  1  mol  of  CO  with  £  mol  O2)  to  take  place  at  a  pressure  p  and 
temperature  2\,  and  heat  the  products  (the  carbon  dioxide)  under  the 
pressure  p  to  the  other  temperature  T2;  and,  in  the  second  process, 
heat  the  reacting  substances  (the  carbon  monoxide  and  oxygen)  under 
the  pressure  p  from  Tl  to  Tv  and  cause  them  to  combine  (forming 
carbon  dioxide)  at  the  pressure  p  and  the  temperature  Tv  Since  in 
each  of  these  processes  the  system  changes  from  the  same  initial  state 
(1  mol  of  CO  and  \  mol  O2  at  p  and  T,)  to  the  same  final  state  (1  mol 
C02  at  p  and  T2),  the  total  change  in  heat-content  must  be  the  same 
in  the  two  processes ;  and  therefore  the  heat-effect  at  constant  pressure 
p  and  temperature  T2  must  differ  from  the  heat-effect  at  constant  pres- 
sure p  and  temperature  T^  by  the  same  amount  as  the  heat  absorbed  in 
heating  the  reaction-products  differs  from  that  absorbed  in  heating  the 
reacting  substances  from  Tt  to  T,  at  the  constant  pressure  p. 


180  THERMOCHEMISTRY 

Pro&.  24>  Demonstrate  the  truth  of  the  principle  stated  in  the  pre- 
ceding paragraph  by  considering  the  increase  in  heat-content  attending 
each  step  of  the  processes  there  described. 

Pro 6.  25.  Calculate  the  heat  of  formation  of  one  formula- weight  of 
PbO  at  200°  from  its  heat  of  formation  ( — 50,300  cal.)  at  20°  and  from 
the  mean  specific  heat-capacities  at  constant  pressure  of  lead  (0.032), 
of  oxygen  (0.212),  and  of  lead  oxide  (0.052). 

Profe.  26.  Calculate  the  heat  of  formation  of  1  mol  gaseous  water  at 
1000°  from  the  following  data.  The  heat  of  formation  of  1  mol  liquid 
water  at  20°  is  —68,400  cal.  Its  heat  of  vaporization  at  100°  is  9670  cal. 
The  molal  heat-capacity  at  constant  pressure  at  T  is  6.50  +  0.0010  T 
for  hydrogen  or  oxygen,  and  8.81  —  0.0019T  -f  0.000,002,221*  for  water. 

Pro&.  27.  Calculate  the  heat  of  the  reaction  K  (liquid)  -f  $Clt  —  KC1 
at  160°,  referring  to  Landolt-Bornstein  Tabellen  for  the  necessary  data. 

Proft.  28.  a.  State  what  quantity  is  equivalent  to  the  change  per 
degree  of  the  heat-effect  of  a  reaction  at  constant  pressure.  6.  Formu- 
late an  integral  which  is  a  general  expression  for  the  difference  In  the 
heat-effects  Q2  and  Qa  of  any  reaction  at  two  temperatures  !F2  and  2\, 
when  no  change  takes  place  in  the  state  of  aggregation  between  those 
temperatures. 


RESULTS  WITH  PERFECT  OASES  131 

THE    RESULTS    OF   THERMOCHEMISTRY 

116.  Constancy  of  the  Heat-Content  of  Perfect  Gases  at  Constant 
Temperature. —  Experiments  have  shown  that  when  a  perfect  gas  ex« 
panels  at  a  constant  temperature  without  producing  any  work  (for 
example,  when  it  expands  within  a  calorimeter  from  one  vessel  into 
another  vessel  previously  evacuated),  there  is  no  heat-effect  in  the 
surroundings  (no  change  of  temperature  in  the  calorimeter).  Such 
experiments  have  established  the  important  law  that  the  energy- 
content  U  of  a  definite  quantity  of  a  perfect  gas  at  any  definite  tem- 
perature has  the  same  value,  whatever  be  its  volume  and  pressure;  in 
other  words,  that  AZ7  — 0  when  a  perfect  gas  changes  its  volume 
and  pressure  at  a  constant  temperature. 

The  sama,, principle  is  evidently  true  of  the  heat-content  H  of  a 
perfect  gas ;  for  this  is  by  definition  equal  to  U  -j-  p  v,  and  p  v  does 
not  change  in  value  when  a  perfect  gas  changes  its  volume  and  pres- 
sure. Therefore,  Aff  =  0  when  a  perfect  gas  changes  its  volume 
and  pressure  at  a  constant  temperature. 

The  law  that  the  energy-content  of  a  perfect  gas  at  a  constant  tem- 
perature is  independent  of  the  volume  leads  to  the  conclusion  that, 
when  the  expansion  of  a  perfect  gas  at  a  constant  temperature  is 
attended  by  the  production  of  work,  there  must  be  a  quantity  of  heat 
absorbed  by  it  equal  to  the  work  produced;  for  it  follows  from  the 
equation  AC7  =  Q  -  -  W  that  Q  =  W  when  AZ7  =  0. 

In  the  case  of  actual  gases  there  are  small  deviations  from  these 
principles  at  moderate  pressures,  and  large  deviations  at  high  pressures, 
in  the  direction  corresponding  to  an  increase  in  energy-content  with 
increase  of  volume.  This  increase  is  most  accurately  determined  by 
experiments,  like  that  described  in  the  following  problem,  in  which  the 
gas  is  caused  to  expand  without  producing  work  (except  that  equiva- 
lent to  the  change  in  its  pressure- volume  product)  and  without  taking 
up  heat  from  the  surroundings.  It  undergoes  thereby  a  decrease  in 
temperature,  which  is  often  called  from  its  discoverers  the  Joule- 
Thomson  Effect.  From  this  decrease  in  temperature  and  from  heat- 
capacity  data  the  quantity  of  heat  which  must  be  imparted  to  the  gas 
to  heat  it  in  its  expanded  state  to  its  original  temperature  is  then 
calculated. 


132  THERMOCHEMISTRY 

ProJ).  29. — Determination  of  the  Change  in  Heat-Content  ~by  Porous- 
Plug  Experiments. — Carbon  dioxide  at  pressure  pl  (e.g.,  2  atin.)  and 
temperature  T^  (e.g.,  20.00°)  is  caused  to  flow  continuously  through  a 
well-insulated  hardwood  tube  containing  a  porous  plug  of  cotton.  On 
passing  through  the  plug  its  pressure  falls  to  p2  (e.g.,  1  atm.),  and  it 
emerges  from  the  tube  at  this  pressure.  After  the  gas  has  flowed  so  long 
that  every  part  of  the  apparatus  has  assumed  the  temperature  of  the 
gas  in  contact  with  it,  the  expansion  of  the  gas  takes  place  without  ex- 
change of  heat  with  the  surroundings.  Its  temperature  after  passing 
through  the  plug  is  found  to  be  T2  (e.  g.,  18.86°).  a.  What  other  process 
must  be  combined  with  this  one  in  order  that  the  net  result  of  the  two 
processes  may  be  the  expansion  of  one  mol  of  the  gas  from  volume  vt 
to  volume  vt  at  a  constant  temperature  Tx?  b.  Formulate  expressions  for 
the  work  produced  W,  the  heat  absorbed  Q,  the  change  in  energy-content 
A  [7,  and  the  change  in  heat-content  A.H  for  each  of  these  two  processes. 
(Note  that  in  the  first  process  a  volume  vt  of  the  gas  disappears  on  one 
side  of  the  plug  under  a  constant  pressure  plt  and  that  a  certain  volume 
17,'  of  the  gas  is  produced  on  the  other  side  of  the  plug  under  a  constant 
pressure  p2.)  G-  Combine  these  results  so  as  to  give  an  expression  for 
the  change  in  energy-content  and  the  change  in  heat-content  that  attends 
a  change  in  volume  from  t^  to  t>2  of  one  mol  of  the  gas  at  a  constant  tem- 
perature Tv  d.  Calculate  in  calories  the  change  in  energy-content  and 
the  change  in  heat-content  attending  the  expansion  of  1  mol  of  carbon 
dioxide  from  a  pressure  of  2  atm.  to  a  pressure  of  1  atm.  at  20°.  Use  the 
following  data  in  addition  to  those  given  above:  the  molal  volume  of 
carbon  dioxide  at  20°  and  2  atm.  is  11,890  ccm.,  and  at  20°  and  1  atm. 
is  23,920  ccm. ;  its  molal  heat-capacity  at  20°  and  at  a  constant  pressure 
of  1  atm.  is  8.92  cal.  per  degree. 

Note. — The  decreases  in  temperature  that  have  been  observed  in 
similar  experiments  with  carbon  dioxide  at  a  series  of  temperatures  are : 
1.35°  at  0°,  1.14°  at  20°,  0.83°  at  60°,  and  0.62°  at  100° ;  and  that 
observed  with  air,  a  more  nearly  perfect  gas,  is  0.25  at  20°.  It  will  be 
noted  that  this  last  value  signifies  that,  when  a  quantity  of  air  expands 
at  20°  from  a  pressure  of  2  atm.  to  one  of  1  atm.,  the  increase  in  its 
heat-content  is  equal  to  the  increase  in  its  heat-content  which  takes 
place  when  it  is  heated  at  constant  pressure  through  0.25°.  This  gives 
a  general  idea  of  the  magnitude  of  the  effect  under  consideration. 

117.  The  Heat-Capacity  of  Perfect  Gases  in  Relation  to  Pressure 
and  Volume. — From  the  law  that  the  heat-content  of  a  perfect  gas 
at  a  definite  temperature  does  not  vary  with  changes  in  its  pressure 
and  volume,  the  following  principles  can  be  derived :  the  heat-capacity 
of  a  perfect  gas  both  at  constant  pressure  and  at  constant  volume  at  a 
definite  temperature  has  the  same  value  whatever  be  the  pressure  and 


RESULTS  WITH  PERFECT  OASES  133 

volume;  and  the  molal  heat-capacity  of  a  perfect  gas  at  constant  pres- 
sure is  greater  than  that  at  constant  volume  by  an  amount  equal  in 
value  to  the  gas-constant  R,  whatever  be  the  gas  and  whatever  be  the 
temperature. 

Prob.  SO.  Derive  the  first  principle  stated  in  the  preceding  text  by 
considering  that  a  perfect  gas  changes  by  two  different  processes  from 
a  volume  v^  at  pressure  PJ  and  temperature  2\  to  a  volume  va  at  pressure 
p2  and  temperature  Tr 

Pro  6.  31.  Derive  the  second  principle  stated  in  the  preceding  text  by 
a  consideration  similar  to  that  employed  in  the  last  problem. 

118.  The  Heat-Capacity  of  Perfect  Gases  in  Relation  to  Compo- 
sition and  Temperature. —  The  molal  heat-capacity  of  gases  at  constant 
volume  depends  primarily  on  the  complexity  of  the  molecules  of  the 
gas.  It  has  the  smallest  value  for  gases  with  monatomic  molecules, 
such  as  mercury  and  helium;  and  it  has  the  same  value,  namely  fR 
or  2.98  calories  per  degree,  for  all  such  gases  at  all  temperatures.  It 
has  a  considerably  larger  value  for  gases  with  diatomic  molecules — 
a  value  which  is  approximately  the  same  for  nearly  all  such  gases, 
namely,  for  H,,  O2,  N2,  NO,  CO,  HC1,  HBr,  and  HI,  and  one  which 
varies  appreciably,  but  not  very  greatly,  with  the  temperature.  Its 
value  for  these  gases  at  any  temperature  T  is  given  by  the  expres- 
sion MVV  =  4.50  -f-  O.OOIOT".  The  corresponding  expressions  for  the 
molal  heat-capacity  of  perfect  gases  at  constant  pressure,  which  has 
been  seen  to  be  always_greater  than  that  at  constant  volume  by  the 
gas-constant  R,  are  MCp  =  4.97  for  monatomic  gases,  and  MGp  = 
6.50  -j-  0.0010  T  for  most  of  the  diatomic  gases.  A  few  diatomic  gases, 
namely,  C12,  Br2,  Ia,  and  IC1,  have  at  room-temperature  larger  values 
of  the  heat-capacity  than  do  the  other  diatomic  gases,  and  the  values 
increase  more  rapidly  with  the  temperature ;  thus,  though  these  heat- 
capacities  have  not  been  satisfactorily  determined,  the  incomplete  data 
that  exist  may  be  expressed  roughly  by  the  equation  MCp=  6.5  + 
0.004  T.  The  only  general  statement  that  can  be  made  in  regard  to 
the  heat-capacities  of  triatomic  and  other  polyatomic  gases  is  that  the 
values  are  much  larger  than  those  for  the  diatomic  gases  and  that  they 
increase  with  the  complexity  of  the  molecule;  thus  the  value  of  MCp 
at  100°  is  6.9  for  17,  and  O2,  8.3  for  H2O  and  HaS,  9.4-9.9  for  CO,, 
S02,  and  N20,  9.0  for  NH.,  21  for  alcohol  (CaH6O),  and  34  for  ether 


134  THERMOCHEMISTRY 

(C4H10O).  In  the  case  of  these  polyatomic  gases  the  increase  of  the 
heat-capacity  with  the  temperature  has  to  be  expressed  by  quadratic 
or  cubic  functions;  thus  in  the  case  of  carbon  dioxide  and  sulphur 
dioxide,  MCP  =  7.0  -f  0.0071T  —  0.00,000,186r2.  The  heat-capacity 
of  water-vapor  can  be  expressed  by  a  similar  function  (see  Prob.  26). 
Prob.  32. — Determination  of  the  Complexity  of  the  Molecules  of  a  Gas 
from  the  Heat-Capacity-Ratio. — From  the  experimentally  determined 
velocity  of  sound  in  a  gas  the  heat-capacity-ratio  CP/CV  can  be  calcu- 
lated. This  ratio  has  been  thus  found  to  be  1.67  in  the  case  of  argon. 
a.  Compare  this  value  with  the  values  of  the  ratio  calculated  for  a 
monatomic  gas  and  for  a  diatomic  gas  at  20°  with  the  aid  of  the  state- 
ments made  in  the  preceding  text.  &.  Show  how  the  atomic  weight  of 
argon  can  be  obtained  by  combining  this  result  with  the  value  of  another 
experimentally  determined  property  of  the  gas. 

119.  Heat-Effects  at  Constant  Pressure  and  at  Constant  Volume 
attending  Reactions  involving  Perfect  Gases. — The  law  that  the  heat- 
content  of  a  perfect  gas  at  a  definite  temperature  is  independent  of  its 
pressure  leads  to  the  conclusion  that  in  the  case  of  reactions  involving 
gases  at  small  pressures  the  heat-effect  Qp  at  constant  pressure  is 
greater  than  the  heat-effect  Qv  at  constant  volume  by  an  amount 
approximately  equal  to  the  work  W  produced  when  the  reaction  takes 
place  at  constant  pressure.  This  work  has  already  been  shown  to  be 
approximately  equal  to  A.ZV  R  T,  where  A7V  denotes  the  increase  in 
the  number  of  mols  of  the  gaseous  substances  present.  The  derivation 
and  application  of  this  principle  are  illustrated  by  the  following 
problems. 

Pro 6.  33.  Show  that  — R  T  is  the  difference  between  the  quantities 
of  heat  absorbed  when  at  T°  a  mixture  of  2  mols  of  CO  and  1  mol  of 
O,  at  1  atm.  unites  to  form  CO2  in  one  case  at  constant  pressure  (for 
example,  in  an  open  calorimeter)  ;  and  in  another  case  at  constant  volume 
(for  example,  in  a  bomb  calorimeter).  Note  that  the  change  of  state 
which  results  when  the  reaction  takes  place  at  a  constant  pressure  of 
1  atm.  could  also  be  brought  about  by  a  process  consisting  of  two  steps, 
namely,  by  causing  the  mixture  of  CO  and  O2  at  1  atm.  to  change  at 
constant  volume  to  CO2  (whereby  the  pressure  would  become  §  atm.), 
and  then  compressing  it  till  its  pressure  becomes  1  atm. 

Prob.  34.  When  1  mol  of  naphthalene  (C10H8)  is  burnt  with  oxygen 
at  20°  in  a  bomb-calorimeter  123,460  cal.  are  evolved.  Calculate  its  heat 
of  combustion  at  constant  pressure. 


HEAT-CAPACITY  OF  SOLID  SUBSTANCES  135 

120.  The  Heat-Capacity  of  Solid  Substances. — Experiments  have 
shown  that  the  atomic  heat-capacity  of  all  solid  elementary  substances 
is  relatively  small  at  temperatures  below  — 100°,  that  with  rising  tem- 
perature it  increases  at  different  rates  in  the  case  of  different  sub- 
stances, and  that,  after  room-temperature  is  reached,  it  increases  as 
a  rule  only  slowly  with  further  increase  of  temperature.  At  room- 
temperature  the  atomic  heat-capacity  has  approximately  the  same  value 
in  the  case  of  all  elements  with  atomic  weights  above  35,  and  in  the 
case  of  the  metallic  elements  of  still  lower  atomic  weight.  The  average 
value  at  room-temperature  is  6.2  calories  per  degree.  Deviations  of 
db  0.4  unit  are  not  uncommon ;  and  deviations  of  +0.7  to  -J-0.9  unit  are 
exhibited  by  some  elements  (for  example,  by  sodium,  potassium,  and 
iodine)  which  at  room-temperature  are  not  much  below  their  melting- 
points.  In  the  case  of  the  non-metallic  elements  with  smaller  atomic 
weights  than  35  the  atomic  heat-capacity  has  a  value  much  smaller 
than  6.2  at  room-temperature;  thus  the  value  for  boron  is  2.6,  for 
graphite  1.9,  for  silicon  4.8,  for  phosphorus  5.6,  and  for  sulphur  5.5. 
This  princip^  in  regard  to  the  approximate  constancy  of  the  atomic 
heat-capacities  of  solid  elementary  substances  is  known  as  the  law  of 
Dulona  and  Petit. 

The  following  table,  which  contains  experimentally  determined 
values  of  the  atomic  heat-capacity,  illustrates  the  preceding  statements : 

Element  At    Wt.   —2^0°      —100°      —50°  0°  50°         10CP         20CP         300° 


Aluminum 

27.1 

2.0 

4.6 

5.3 

5.6 

5.8 

6.0 

6.3 

6.7 

Antimony 

120.2 

4.2 

5.4 

5.7 

5.9 

6.0 

6.1 

6.2 

6.3 

Cadmium 

112.4 

4.3 

RjR 

6.1 

6.2 

6.2 

6.3 

6.8 

9.4 

Chromium 

52.0 

2.0 

4.1 

4.9 

5.4 

5.7 

5.8 

6.1 

6.4 

Copper 

F3.6 

3.3 

5.0 

5.5 

5.8 

5.9 

6.0 

6.1 

6.2 

Iron 

55.8 

3.0 

4.6 

5.2 

5.8 

6.2 

6.6 

7.0 

7.4 

Lead 

207.1 

5.6 

5.0 

6.0 

6.2 

6.3 

6.5 

6.7 

7.0 

Magnesium 

24.3 

3.5 

4.9 

5.4 

5.8 

6.0 

6.2 

6.9 

7.7 

Silver 

107.9 

4.0 

5.6 

6.0 

6.3 

6.5 

6.6 

6.9 

7.1 

Sodium 

23.0 

5.1 

5.0 

6.3 

6.7 

7.1 

7.5 

•• 

•• 

Boron 

11.0 

. 

1.6 

2.0 

2.4 

2.8 

3.2 

4.0 

.. 

Diamond 

12.0 

0.1 

0.5 

0.8 

1.1 

1.6 

2.3 

3.2 

3.9 

Sulphur 

32.1 

2.5 

4.3 

4.9 

5.3 

5.6 

5.8 

,  . 

.  . 

The  law  of  Dulong  and  Petit,  even  though  it  is  only  an  approxi- 
mate principle,  may  evidently  be  employed  for  determining  what 
multiple  of  the  combining  weight  of  an  element  is  its  atomic  weight; 


136  THERMOCHEMISTRY 

and  the  application  of  this  law  was  in  fact  one  of  the  most  important 
methods  by  which  the  present  system  of  atomic- weight  values  was 
originally  established. 

A  simple  principle  has  also  been  discovered  in  regard  to  the  formal 
heat-capacity  of  solid  compound  substances  at  room-temperature.  It 
has  been  found,  namely,  that  this  property  is  approximately  an  additive 
one,  that  is,  one  whose  value  can  be  approximately  calculated  by  add- 
ing together  certain  values  representing  the  heat-capacity  of  the  ele- 
ments contained  in  the  compound.  This  principle  is  expressed  by  the 
following  equation,  which  shows  at  the  same  time  the  values  of 
the  constants  (the  so-called  atomic  heat-capacities)  for  all  the  common 
elements : 

MCP  =  6.2  TIE  +  4.0  no  +  2.3  tin  +  1.8  nc  +  5.4  ns  +  2.7  n& 

+  5.4  nP  +  3.8  nSi. 

In  tljis  equation  MCp  represents  the  formal  heat-capacity  of  the  com- 
pound at  constant  pressure  at  room-temperature;  no,  nu,  nc,  n& 
KB*  nP>  and  ns\  are  the  number  of  atomic  weights  of  oxygen,  hydro- 
gen, carbon,  sulphur,  boron,  phosphorus,  and  silicon  present  in  one 
formula-weight  of  the  compound;  and  ??E  is  the  number  of  atomic 
weights  of  any  other  element  so  present.  The  values  given  for  the 
constants  are  average  values  derived  from  heat-capacity  measurements 
with  solid  compounds. 

The  following  table  illustrates  the  degree  of  correspondence  which 
exists  between  the  values  of  the  formal  heat-capacity  so  calculated  and 
those  measured  experimentally : 


H20  (ice) 
A130,     . 


Cole. 

Meas. 

Substance 

Cole. 

Meas. 

8.6 

9.7 

PbN2O6 

42.6 

38.8 

24.4 

20.5 

CaSiO8 

22.0 

21.3 

24.4 

25.6 

K4Fe(CN)fl 

79.0 

78.8 

28.6 

28.7 

CuSO4.5H2O 

70.6 

67.2 

12.4 

12.4 

A1K(S04)2.12H20 

158. 

165. 

18.6 

18.5 

C10H8 

36. 

40. 

20.0 

20.2 

H2C204 

24.2 

25.1 

Sb,S, 
KC1 
PbCl, 
CaCO, 

It  will  be  observed  that  differences  of  ten  percent  between  the  calcu- 
lated and  measured  values  are  not  uncommon. 

Pro&.  35.  Calculate  an  approximate  value  at  20°  of  the  specific  heat- 
capacity  at  constant  pressure  of  a,  platinum  ;  &,  silver  bromide  ;  c,  potas- 
sium sulphate  ;  d,  benzophenone,  C^H^O.  Find  the  percentage  deviations 


CHANGES  IN  STATE  OF  AGGREGATION  137 

of  these  values  from  the  measured  values,  which  are,  a,  0.032  ;  &,  0.074  ; 
c,  0.190  ;  and  d,  0.31. 

Prob.  36.  —  Determination  of  Atomic  Weights  from  Heat-Capacity 
Measurements.  —  Calculate  the  exact  atomic  weight  of  an  element  whose 
specific  heat-capacity  is  0.092,  and  whose  oxide  contains  88.82%  of  the 
element. 

121.  Heat-Effects  attending  Changes  in  the  State  of  Aggregation 
of  Substances.  —  The  heat  of  vaporization  of  liquid  substances,  the 
heat  of  fusion  of  solid  substances,  and  the  heat  of  transition  of  one 
solid  substance  into  another  (as  of  rhombic  into  monoclinic  sulphur) 
are  quantities  which  are  important  in  themselves  and  which  are  fre- 
quently involved  in  calculations  of  the  heat  of  chemical  reactions. 
The  general  statement  can  be  made  in  regard  to  them  that  the  conver- 
sion of  the  form  that  is  stable  at  lower  temperatures  into  that  stable 
at  higher  temperatures  (for  example,  of  ice  into  water,  or  of  rhombic 
into  monoclinic  sulphur)  is  always  attended  by  a  positive  heat-effect, 
that  is,  by  an  absorption  of  heat. 

The  following  simple  principle,  known  as  Trouton's  rule,  has  been 
discovered  in  regard  to  the  values  of  the  heat  of  vaporization:  the 
ratio  of  the  molal  heat  of  vaporization  of  a  liquid  at  its  boiling-point 
to  its  boiling-point  on  the  absolute  scale  has  approximately  the  same 
value  (namely,  about  20.5)  for  all  liquids  except  those  whose  molecules 
are  associated;  that  is,  M  L/T  =  approx.  20.5.  The  actual  values  of 
these  quantities  in  the  case  of  five  very  different  liquids  are  shown 
in  the  following  table  : 


Substance  ML  T 

Bromine  67GO  332  20.4 

Benzene  7350  353  20.8 

Carbon  bisulphide  6380  319  20.0 

Ethyl  ether  6260  308  20.3 

Ethyl  formate  7J80  327  22.0 

Substances  containing  the  hydroxyl  group,  such  as  water,  alcohols, 
and  acids,  whose  molecules  in  the  liquid  state  are  for  other  reasons 
believed  to  be  associated  (see  Art.  26,  Note  at  end),  form  marked 
exceptions  to  Trouton's  rule.  Thus  the  value  of  M~L/T  is  25.9  for 
water,  27.0  for  ethyl  alcohol,  and  14.9  for  acetic  acid. 

The  molal  heat  of  vaporization  of  a  liquid  or  solid  substance  can 
also  be  calculated  by  the  approximate  form  of  the  Clausius'  equation 
(Art.  22)  from  the  change  of  its  vapor-pressure  with  the  temperature 


138  THERMOCHEMISTRY 

The  heat  of  solution  of  substances  is  another  important  quantity, 
In  determining  and  expressing  it  the  quantity  of  solvent  in  which  a 
definite  weight  of  the  substance  is  dissolved  must  be  taken  into  con- 
sideration. The  two  limiting  cases  are  the  heats  of  solution  in  a  very 
large  quantity  of  solvent  and  in  that  quantity  of  solvent  which  forms 
with  the  substance  a  saturated  solution.  These  two  heat-effects  some- 
times have  different  signs.  They  evidently  differ  by  the  heat  of  dilu- 
tion of  the  saturated  solution  with  a  large  quantity  of  water. 

The  dissolving  of  gaseous  substances  in  solvents  is  always,  and  that 
of  liquid  substances  is  usually,  attended  by  an  evolution  of  heat;  and 
the  dissolving  of  solid  substances  in  solvents  is  usually  attended  by 
an  absorption  of  heat. 

The  heat  of  dilution  of  substances  in  solution  is  also  important; 
for  it  enables  the  heat  of  formation  of  a  solution  of  one  concentration 
to  be  calculated  from  that  of  a  solution  of  another  concentration,  and 
thus  enables  heats  of  reaction  to  be  calculated  at  different  concentra- 
tions. The  heat-effect  attending  the  addition  of  an  equal  volume  of 
water  to  a  concentrated  aqueous  solution  is  often  large ;  but  it  becomes 
less  as  the  concentration  diminishes;  and,  after  a  moderately  small 
concentration  (such  as  0.2  formal)  has  been  attained,  there  is  usually 
only  a  very  small  heat-effect  on  adding  even  a  very  large  quantity  of 
water.  For  example,  on  adding  at  18°  to  1  formula- weight  of  gaseous 
HC1  or  of  solid  ZnCl2  successively  AAT  formula-weights  of  water,  there 
are  evolved  the  following  quantities  of  heat  ( — Q}  in  calories: 

Atf 5  5  10  30          50       100       200 

—0  for  HC1...         14960        1200          600          360        120         50 

— QforZnCl,..          7740        1850        1300        2170      1490       820       390 

Proft.  57.  The  heat  of  fusion  of  1  g.  of  ice  at  0°  is  79.7  cal.  What 
other  data  would  be  needed  to  calculate  the  heat-effect  attending  the 
melting  of  1  g.  of  ice  into  1000  g.  of  a  normal  solution  of  sodium  chloride 
at  its  freezing-point,  — 3.42°,  and  how  would  the  calculation  be  made? 

Prob.  38.  The  heat  of  solution  at  20°  in  a  large  quantity  of  chloro- 
form of  1  at.  wt.  rhombic  sulphur  is  -j-640  cal.  and  of  1  at.  wt.  monoclinic 
sulphur  is  -(-560  cal.  Show  by  the  law  of  initial  and  final  states  what 
other  heat-effect  can  be  derived  from  these  data,  and  what  its  value  is. 

Prob.  39. — Change  of  Heat  of  Reaction  with  the  Concentration. — 
The  heat  of  solution  at  18°  of  IZn  in  HC1  200Aq  is  —34,200  cal.  Find  its 
heat  of  solution  in  HC1 5Aq,  a,  by  applying  the  law  of  initial  and  final 
states,  and  &,  by  formulating  the  thermochemical  equations  involved. 


REACTIONS  IN  AQUEOUS  SOLUTION  139 

There  is  another  kind  of  heat  of  dilution  or  solution  that  is  of  great 
importance  in  the  thermodynamic  treatment  of  electromotive  force 
and  of  chemical  equilibrium  (considered  in  Chapters  VIII  and  IX). 
This  is  the  heat-effect  attending  the  addition  of  one  formula-weight 
of  the  solvent  or  solute  to  an"  infinite  quantity  of  the  solution.  The 
value  of  this  quantity,  which  is  known  as  the  partial  heat  of  dilution 
or  solution,  can  be  derived  from  actual  data  like  those  given  in  the 
preceding  table  in  the  ways  illustrated  by  the  following  problems. 

Pro  6.  40.  Determine  with  the  aid  of  a  plot  the  heat-effect  attending 
the  addition  of  1H2O  to  an  infinite  quantity  of  ZnCl2100HsO  at  18°. 

Prob.  41.  The  measured  heat-effect  Q  attending  the  mixing  of 
1H2SO4  with  ;VH2O  at  18°  is  expressed  by  the  empirical  equation, 
Q  =  —17860  N/(N  Jf-  1.80)  for  values  of  N  not  exceeding  20.  Calculate 
the  heat-effect  attending  the  addition  of  1H2O  to  an  infinite  quantity  of 
HaSO410H2O. 

Prob.  42.  Derive  a  relation  between  the  heat-effect  attending  the 
dissolving  of  !ZnCl2  in  100H2O,  that  attending  the  addition  of  100H2O 
to  an  infinite  quantity  of  a  solution  of  the  composition  !ZnCl2100HaO,  and 
that  attending  the  addition  of  IZnCl,  to  an  infinite  quantity  of  such  a 
solution. 

Prob.  43.    Calculate  the  heat-effect  attending  the  addition  of: 
a,  !ZnCl2  to  an  infinite  quantity  of  lZnC!2100H2O. 
6,  1H2SO4  to  an  infinite  quantity  of  1H2SO410H2O. 

Prob.  44.  a.  The  heat-content  of  !ZnCl2  is  — 97,300  cal.  Calculate 
the  heat-content  of  !ZnCl2  in  !ZnCl2100H2O  (by  which  is  meant  the 
increase  in  heat-content  in  starting  with  IZn,  1C72,  and  an  infinite  quan- 
tity of  a  solution  of  the  composition  ZnCl2100H2O,  and  ending  with 
IZnClj  in  that  solution),  b.  Give  the  values  of  the  other  symbols  in  the 
thermochemical  equations : 

ZnCl2100H2O  =  ZnCl2  (in  ZnCl2100H2O )  -f  100H2O  (in  ZnCl2100H3O )  -f  0  cal. 
ZnCl2100Aq  =  ZnCl2(in  ZnCl2100H2O)  -f  100Aq(in  ZnCljlOOI^O)  -f  0  cal. 

122.  Heats  of  Reaction  in  Aqueous  Solution. — The  investigations 
made  of  the  heat-effects  attending  chemical  reactions  in  aqueous 
solution  between  substances  present  at  fairly  small  concentrations 
(0.1-0.3  normal)  have  established  the  following  principles : 

1.  On  mixing  solutions  of  two  neutral  salts  which  do  not  form  a 
precipitate  by  metathesis  (for  example,  solutions  of  potassium  chloride 
and  sodium  sulphate)  there  is  scarcely  any  heat-effect.  Exceptions  to 
this  principle  are  met  with  in  the  few  cases  in  which  an  unionized 


140  THERMOCHEMISTRY 

salt  is  produced  by  the  metathesis.  Thus  the  metathetical  reaction 
2K+C1-  +  Hg*+(NO8-)2  =  2K*N03-  +  HgCl2  is  attended  by  a  heat- 
effect  of  —12,400  cal. 

2.  The  heat  of  neutralization-  of  a  solution  of  any  largely  ionized 
monobasic  acid  with  a  solution  of  any  largely  ionized  monacidic  base 
(for   example,   of  hydrochloric   acid,   nitric   acid,   etc.,   with   sodium 
hydroxide,  potassium  hydroxide,   etc.)    has  approximately  the  same 
value,  whatever  be  the  acid  or  base.    At  18°  this  nearly  constant  value 
averages  — 13,810  cal.  per  equivalent  when  the  acid  and  base  solutions 
are  0.12  to  0.25  normal. 

3.  When  the  base  or  acid  is  only  partly  ionized  (as  in  the  case  of 
ammonium  hydroxide  or  hydrofluoric  acid),  the  heat-effect  attending 
its  neutralization  with  a  largely  ionized  acid  or  base  is  often  much 
larger  or  smaller  than  that  observed  when  both  acid  and  base  are 
largely  ionized;  thus  the  heat  of  neutralization  of  one  equivalent  of 
ammonium  hydroxide  with  one  of  hydrochloric  acid  is  — 12,300  cal., 
and  that  of  one  equivalent  of  hydrofluoric  acid  with  one  of  sodium 
hydroxide  is  —16,300  cal. 

4.  When  one  formula-weight  of  a  dibasic  acid  is  neutralized  in 
steps  by  adding  first  one  equivalent  of  a  largely  ionized  base  and  then 
a  second  equivalent,  the  heat-effects  for  the  two  equivalents  of  base 
are  usually  different;  for  example,  at  18°   the  two  heat-effects  are 
— 14,600  and  — 16,600  cal.  in  neutralizing  0.28  normal  sulphuric  acid 
with  0.28  normal  sodium  hydroxide,  and  they  are  — 11,100  and  — 9,100 
cal.  in  neutralizing  carbonic  acid  with  that  base. 

5.  When  certain  polybasic  acids  are  neutralized,  there  is  sometimes 
scarcely  any  heat-effect  when  the  second  or  third  equivalent  of  base 
is  added.     Thus  the  successive  heat-effects  when   phosphorous   acid 
(H,POS)  is  treated  with  sodium  hydroxide  at  18°  are:  —14,800  cal. 
with  the  first  equivalent,  — 13,600  cal.  with  the  second  equivalent,  and 
only  — 500  cal.  with  the  third  equivalent ;  and  with  hypophosphorous 
acid  (H,POa)  there  is  a  heat-effect  of  —15,200  cal.  with  the  first  equiva- 
lent   of    sodium    hydroxide,    and    only   — 110  cal.    with    the    second 
equivalent. 

6.  With  certain  polybasic  acids  there  is  a  considerable  heat-effect 
when  to  the  solution  of  the  neutral  salt  another  equivalent  of  base  is 
added;  thus,  there  is  a  heat-effect  of  — 1200  cal.  on  mixing  a  solution 


REACTIONS  IN  AQUEOUS  SOLUTION  141 

containing  one  equivalent  of  sodium  hydroxide  with  one  containing 
one  formula-weight  of  sodium  phosphate  (Na3PO4). 

Frol.  45.  a.  Explain  the  first  principle  stated  in  the  preceding  text 
with  the  aid  of  the  ionic  theory,  assuming  that  the  solutions  are  very 
dilute.  6.  What  conclusion  as  to  the  heat  of  ionization  of  neutral  salts 
can  be  drawn  from  the  fact  that  this  principle  holds  true  even  at  fairly 
high  concentrations  (e.  g.,  0.3  normal)?  c.  Write  a  thermochemical 
equation  corresponding  to  the  ionic  reaction  to  which  the  heat-effect  is 
mainly  due  in  the  reaction  cited  as  an  exception  to  the  principle. 

Pro&.  46.  Show  that  on  mixing  solutions  of  two  salts  (such  as  lead 
nitrate  and  potassium  iodide)  which  form  a  precipitate  by  metathesis 
there  must  be  a  heat-effect  which  is  substantially  equal,  but  opposite  in 
sign,  to  the  heat  of  solution  of  the  precipitated  substance  (the  lead 
iodide) . 

Pro&.  47.  a.  Write  the  ionic  reaction  to  which  the  nearly  constant 
heat  of  neutralization  of  largely  ionized  acids  and  bases  corresponds. 
&.  What  other  heat-effect  is  involved  in  the  neutralization  of  ammonium 
hydroxide  (a  slightly  ionized  base)  with  a  largely  ionized  acid,  and  what 
is  its  value?  c.  What  is  the  heat-effect  that  attends  the  reaction  between 
1NH4C1  and  INaOH  in  0.2  normal  solution? 

Prol).  48.  Calculate  the  heat  of  ionization  of  one  formula-weight  of 
HF  from  the  facts  that  its  heat  of  neutralization  in  0.28  normal  solution 
with  0.28  normal  NaOH  solution  has  been  found  to  be  — 16,300  cal.,  and 
its  ionization  in  0.28  normal  solution  is  estimated  to  be  5.2%. 

Pro&.  49.  Calculate  the  heats  of  ionization  at  18°  that  can  be  derived 
from  the  heats  of  neutralization  of  carbonic  acid  given  in  the  text. 
Carbonic  acid  is  a  very  slightly  ionized  acid ;  sodium  hydrogen  carbonate 
solution  is  practically  neutral;  and  the  sodium  carbonate  in  the  0.07 
formal  solution  produced  by  the  neutralization  is  6.3%  hydrolyzed. 

Pro 6.  50.  Conductance,  transference,  and  reaction-rate  measurements 
have  shown  that  sodium  hydrogen  sulphate  in  0.1  formal  solution  at  18° 
consists  approximately  of  8%  NaHSO4, 12%  Na2SO4,  44%  HSO4-,  36%-SO4=, 
and  of  the  corresponding  amount  of  Na+  and  H+.  A  calorimetric 
measurement  at  18°  has  given  the  result  expressed  by  the  equation : 

NaHS04601Aq  +  NaOH200Aq  =  Na2SO4801Aq  -|_  H2O  +  16,620  cal. 
Calculate  the  heat  of  the  reaction  HSO4-  =  H+  4.  SO4=,   assuming  that 
the  heats  of  ionization  of  Na2SO4  and  of  NaHSO4  (into  Na*  and  HSO4-) 
are  zero. 

Pro  6.  51.  What  do  the  heat-effects  observed  in  the  neutralization  of 
phosphorous  acid  and  of  hypophosphorous  acid  show  as  to  the  existence 
of  salts  of  these  acids  in  solution? 

Pro&.  52.  Explain  the  fact  stated  in  the  last  paragraph  of  the  pre- 
ceding text. 


142  THERMOCHEMISTRY 

123.   Applications  of  Thermochemical  Principles. 

Determination  of  the  Change  of  the  Heat  of  Reaction  with  the 
Temperature. — 

Prob.  53.  Derive  a  numerical  expression  for  the  heat  of  ionization 
of  water  as  a  function  of  the  temperature  from  the  following  data.  The 
heat  evolved  on  mixing  at  18°  a  solution  of  INaOH  in  200H2O  with  one 
of  1HC1  in  200H,O  is  13,810  cal.  The  specific  heat-capacity  of  the  sodium 
hydroxide  solution  is  0.9827,  that  of  the  hydrochloric  acid  solution  is 
0.9814,  and  that  of  the  sodium  chloride  solution  produced  is  0.9887.  (Tt 
will  be  noted  that  the  heat-capacity  data  must  be  known  with  great 
accuracy  in  the  case  of  reactions  between  solutes  in  dilute  solution.) 

Prob.  54.  The  heat  of  formation  of  1  mol  of  gaseous  HI  at  18°  is 
4-6200  cal.  Estimate  the  value  of  its  heat  of  formation  at  300°  with 
the  aid  of  the  principles  relating  to  the  values  of  heat  quantities  stated 
in  this  chapter,  and  from  the  fact  that  the  vapor-pressure  of  solid  iodine 
is  70  mm.  at  109°  and  89  mm.  at  its  melting-point,  114°. 

Prob.  55. — Heat  Evolved  by  Continuous  Processes  at  High  Temper- 
atures.— A  mixture  of  oxygen  and  hydrogen  chloride  in  the  proportion 
$O, :  1HC1  at  18°  is  passed  continuously  into  a  vessel  at  386°  containing 
a  suitable  catalyzer.  The  gas  is  passed  so  slowly  that  equilibrium  is 
established,  80%  of  the  hydrogen  chloride  being  converted  into  chlorine 
and  water.  Calculate  the  heat  that  will  be  actually  given  off  from  the 
equilibrium  vessel  per  mol  of  HC1  passed  through. 

Maximum  Temperature  Producible  by  Chemical  Changes. — 

Prob.  56.  Calculate  the  maximum  temperature  that  could  theoret- 
ically be  attained  in  the  flame  produced  by  burning  at  18°  a  "water-gas" 
consisting  of  equimolal  quantities  of  hydrogen  and  carbon  monoxide 
with  twice  the  quantity  of  air  required  for  complete  combustion.  Assume 
that  there  is  no  loss  of  heat  to  the  surroundings  and  that  the  reaction- 
products  are  not  appreciably  dissociated. 

Prob.  57.  a.  Calculate  the  maximum  temperature  and  pressure  that 
could  be  produced  by  the  explosion  within  a  bomb  of  a  mixture  consist- 
ing of  1  mol  H2,  \  mol  O2,  and  1  mol  N2,  at  18°  and  100  mm.,  assuming 
that  the  water  produced  is  not  appreciably  dissociated,  b.  The  maxi- 
mum pressure  produced  in  an  actual  experiment  was  found  to  be  840  mm. 
Show  how  from  this  result  the  degree  of  dissociation  of  the  water- vapor 
and  the  temperature  of  the  mixture  at  the  moment  of  the  explosion  can 
be  calculated.  (The  equations  should  be  formulated,  but  they  need  not  be 
solved  numerically. ) 

Prob.  58. — Determination  of  Chemical  Equilibria  by  Thermochemical 
Methods. — Describe  a  thermochemical  method  of  determining  the  extent 
to  which  acetic  acid  displaces  hydrofluoric  acid  from  sodium  fluoride  in 
dilute  solution.  (The  heat  of  neutralization  of  acetic  acid  with  sodium 
hydroxide  at  18°  is  —13,230  cal.) 


CHAPTEK   IX 

ELECTROCHEMISTRY:  THE  PRODUCTION   OF  ELECTRI- 
CAL ENERGY  BY  CHEMICAL  CHANGES  AND  OF 
CHEMICAL  CHANGES  BY  ELECTRICAL  ENERGY 


THE   SECOND  LAW  OF  ENERGETICS 

124.  The  Second  Law  of  Energetics. — Before  the  production  of 
electrical  energy  by  chemical  changes  can  be  adequately  considered, 
familiarity  with  certain  aspects  of  another  general  principle  relating 
to  energy,  the  so-called  second  law  of  energetics,  is  essential. 

The  first  law  of  energetics  states  that  when  one  form  of  e.nergy  is 
converted  into  another  the  quantity  of  the  form  of  energy  that  is  pro- 
duced is  equivalent  to  the  quantity  of  the  form  that  disappears;  but 
it  does  not  indicate  that  there  is  any  other  restriction  as  to  the  trans- 
formability  of  the  different  forms  of  energy,  Experience  has  shown, 
however,  that  while  the  various  forms  of  work  can  be  completely 
transformed  into  one  another  and  into  heat,  the  transformation  of  heat 
into  work  is  subject  to  certain  limitations.  Thus,  no  system  (com- 
bination of  matter)  has  ever  been  discovered  which,  as  a  result  of  any 
process  taking  place  continuously  in  it,  can  produce  work  in  unlimited 
quantity  merely  by  withdrawing  heat  from  the  surroundings.  An 
ideal  process  which  is  conceived  to  produce  this  result  may  be  called 
perpetual  motion  of  the  second  kind;  and  the  experience  just  men- 
tioned may  be  expressed  by  the  statement  that  perpetual  motion  of 
the  second  kind  is  impossible.  This  is  the  perpetual-motion  principle 
that  was  stated  and  illustrated  in  Art.  28.  Perpetual  motion  of  the 
second  kind,  by  which  work  is  conceived  to  be  produced  out  of  heat, 
is  to  be  distinguished  from  perpetual  motion  of  the  first  kind  (described 
in  Art.  107),  by  which  work  is  conceived  to  be  produced  without  con- 
suming energy  of  any  kind. 

Experience  with  processes  taking  place  at  different  temperatures 
has  led  to  the  conclusion  that  this  principle  is  a  consequence  of  a 
drill  more  general  law,  known  as  the  second  law  of  energetics,  which 
may  be  expressed  as  follows:  a  process  whose  final  result  is  only  a 
transformation  of  a  quantity  of  heat  into  work  is  an  impossibility. 

143 


144  ELECTROCHEMISTRY 

Prob.  1.  a.  State  what  energy-effects  occur  when  a  perfect  gas  ex- 
pands at  a  constant  temperature  against  an  external  pressure.  6.  Explain 
why  this  is  not  a  contradiction  of  the  second  law  of  energetics. 

This  law  does  not  imply  that  heat  cannot  be  transformed  into 
work,  but  only  that  its  transformation  must  be  attended  by  some  other 
change.  This  attendant  change  may  consist  in  a  permanent  change 
in  the  state  of  the  system  by  which  the  energy-transformation  is 
brought  about;  or,  when  the  system  undergoes  no  permanent  change 
in  state,  it  consists  in  the  passage  of  an  additional  quantity  of  heat 
from  a  higher  to  a  lower  temperature.  The  latter  effect,  which  can 
take  place  only  when  there  is  difference  of  temperature  in  different 
parts  of  the  surroundings,  will  be  considered  later.  In  this  chapter 
will  be  considered  only  the  production  of  work  by  changes  in  state 
taking  place  in  surroundings  of  constant  temperature. 

When  the  process  by  which  any  change  in  state  takes  place  is  one 
that  produces  a  quantity  of  work  equal  to  (or  differing  by  only  an 
infinitesimal  amount  from)  the  quantity  of  work  which  must  be  ex- 
pended in  order  to  restore  the  system  to  its  original  state,  the  process 
is  called  a  reversible  process.  When  the  process  is  one  that  produces 
a  smaller  quantity  of  work  than  that  which  must  be  expended  in 
restoring  the  system  to  its  original  state,  it  is  called  an  irreversible 
process.  Similarly,  when  work  has  to  be  expended  in  changing  the 
state  of  a  system,  the  process  is  said  to  be  reversible  when  the  amount 
of  work  expended  is  equal  to  that  which  can  be  produced  when  the 
change  in  state  takes  place  in  the  opposite  direction,  and  to  be 
irreversible  when  the  amount  of  work  expended  is  larger  than  that 
which  can  be  so  produced. 

It  is  to  be  noted  that  the  term  reversible  is  always  employed,  in 
the  manner  just  defined,  to  designate  a  process  of  such  a  character 
that  it  is  possible  to  restore  the  original  condition  of  things  both  in 
the  system  and  its  surroundings.  After  an  irreversible  process  has 
taken  place,  it  is  in  general  possible  to  restore  the  system  to  its 
original  state,  but  only  by  withdrawing  from  the  surroundings  a 
larger  quantity  of  work  than  was  produced  in  them,  so  that  the 
original  condition  in  the  surroundings  is  not  reproduced.  When  an 
irreversible  change  has  once  taken  place,  it  is  not  possible  by  any 
means  whatever  to  reproduce  in  their  entirety  the  conditions  that 
previously  existed. 


THE  SECOND  LAW  OF  ENERGETICS  145 

Prol).  2.  A7  mols  of  a  perfect  gas  having  a  pressure  of  2  atm.  are 
enclosed  within  a  cylinder  placed  in  a  thermostat  at  a  temperature  T 
and  provided  with  a  weighted  piston.  The  weight  on  the  piston  is 
suddenly  reduced^so  that  it  exerts  on  the  gas  a  pressure  of  1  atm.,  and 
the  gas  expands  till  its  own  pressure  becomes  1  atm.  Explain  why  this 
process  is  irreversible;  and  state  what  would  have  to  be  true  of  the 
pressure  exerted  by  the  piston  on  the  gas  during  its  expansion  in  order 
that  the  process  might  be  reversible. 

Prol).  8.  a.  Derive  an  expression  in  terms  of  A7,  R,  and  T  for  the  work 
produced  by  the  irreversible  process  described  in  Prob.  2.  ft.  Derive  a 
corresponding  expression  for  the  work  produced  when  the  same  change 
in  state  is  brought  about  by  a  reversible  process,  c.  Find  the  numerical 
ratio  of  these  two  quantities  of  work. 

Prol).  4.  Two  Daniell  cells,  each  having  an  electromotive  force  of 
1.10  volts,  are  connected  in  series  and  are  used  for  charging  a  lead 
storage-cell  having  a  (counter)  electromotive  force  of  2.10  volts.  Ex- 
plain why  the  process  is  not  reversible;  and  state  how  a  number  of 
Daniell  cells  and  a  number  of  storage-cells  could  be  so  arranged  that 
the  latter  might  be  charged  reversibly.  (In  this  case  the  Daniell  cells 
may  be  regarded  as  the  system,  and  the  storage  cells  as  a  part  of  the 
surroundings  in  which  the  electrical  work  is  produced  and  stored.) 

As  illustrated  by  the  preceding  problems,  in  order  that  the  process 
by  which  a  change  in  state  is  brought  about  may  be  reversible,  the 
pressure  externally  applied  must  be  substantially  equal  to  the  pressure 
exerted  by  the  system  itself,  or  the  applied  electromotive  force  must 
be  substantially  equal  to  the  electromotive  force  of  the  cell.  For  the 
change  in  state  will  take  place  in  one  direction  when  the  applied  pres- 
sure or  electromotive  force  is  only  infinitesimally  less  than  that  of 
the  system,  and  in  the  other  direction  when  it  is  only  infinitesimally 
greater;  but,  if  the  applied  pressure  or  electromotive  force  were  less 
by  a  finite  amount  than  that  of  the  system,  the  .quantity  of  mechanical 
or  electrical  work  produced  in  the  surroundings  would  evidently  not 
suffice  to  restore  the  system  to  its  initial  state;  and  if  the  applied 
pressure  or  electromotive  force  were  greater  than  that  of  the  system, 
more  work  would  be  withdrawn  from  the  surroundings  than  the  system 
would  be  capable  of  reproducing  on  reverting  to  its  original  state. 

125.  Application  of  the  Second  Law  to  Isothermal  Changes  in 
State.  The  Concept  of  Free  Energy. 

Prol).  5:  In  a  voltaic  cell  consisting  of  one  platinum  electrode  in 
contact  with  a  0.1  normal  HC1  solution  and  with  hydrogen  gas  at  1  atm. 
and  of  a  second  platinum  electrode  in  contact  with  the  same  HC1  solution 


146  ELECTROCHEMISTRY 

and  with  hydrogen  gas  at  0.1  atm.,  hydrogen  is  found  to  go  into  solu- 
tion (as  hydrogen-ion)  at  the  first  electrode  and  to  be  evolved  at  the 
second  electrode,  as  a  result  of  the  electromotive  force  which  is  produced. 
o.  Name  the  change  in  state  that  takes  place  in  such  a  cell  when  one 
faraday  of  electricity  passes  through  it  at  18°,  specifying  all  the  factors 
determining  the  change  in  state  (see  Art.  108,  first  paragraph),  but  dis- 
regarding the  transference  in  the  solution,  which  in  a  cell  of  this  kind 
will  be  shown  later  to  be  attended  by  no  energy-effect.  6.  Describe  how 
the  same  change  in  state  could  be  brought  about  reversibly  by  a  process 
not  involving  voltaic  action,  c.  Show  by  the  perpetual-motion  principle 
that  the  quantity  of  work  attending  this  process  is  equal  to  the  work 
produced  when  the  same  change  in  state  takes  place  reversibly  in  the  cell. 

The  principle,  illustrated  by  the  preceding  problem,  that  the 
quantity  of  work  produced  or  expended  when  a  definite  change  in  the 
state  of  a  system  takes  place  at  a  constant  temperature  by  a  reversible 
process  is  the  same  whatever  be  the  nature  of  the  reversible  process, 
is  of  great  importance  in  chemical  considerations.  This  principle 
shows  that  any  system  in  any  definite  state  has  a  certain  power  of 
producing  work,  and  that  this  power  changes  by  a  definite  amount 
when  a  definite  change  in  state  takes  place.  The  power  of  producing 
work  is  therefore,  like  the  energy-content  and  the  heat-content,  a  quan- 
tity determined  by  the  state  of  the  system;  and,  in  analogy  with  the 
names  given  to  them,  it  may  be  called  the  work-content  (A}  of  the 
system.  The  absolute  value  of  this  quantity  cannot  be  determined; 
but  the  change  in  its  value  when  any  definite  change  in  state  takes 
place  at  a  constant  temperature  is  measured  by  the  work  (TFR) 
produced  when  the  change  in  state  takes  place  reversibly.  Thus, 
representing  by  Al  the  work-content  of  the  system  in  the  initial 
state,  and  by  A2  that  in  the  final  state,  the  decrease  in  work-content 
4  —  A,  =  —  Al  =  WR. 

It  will  be  noted  that,  while  the  First  Law  requires  that  there  be 
a  quantity  of  energy  produced  in  the  surroundings  equal  to  the  decrease 
of  the  energy-content  of  the  system,  the  Second  Law  does  not  require 
that  there  be  a  quantity  of  work  (W)  produced  equal  to  the  decrease 
in  work-content  (as  was  illustrated  by  Prob.  3).  The  Second  Law 
requires  only  that  the  quantity  of  work  produced  be  not  greater  than 
the  decrease  in  work-content.  That  is,  W  >  —  AA  for  no  process 
whatever;  W  =  —  A  A  for  a  reversible  process;  and  W  <  —  A  A  for 
an  irreversible  one. 


THE  SECOND  LAW  OF  ENERGETICS  14? 

Just  as  it  is  more  convenient  in  chemical  considerations  to  con- 
sider the  heat-content  rather  than  the  energy-content  of -systems,  so 
there  are  many  advantages  in  considering  in  place  of  the  work-content 
a  quantity  which  differs  from  it,  just  as  the  heat-content  differs  from 
the  energy-content,  by  the  value  of  the  pressure-volume  product.  This 
quantity,  which  may  be  called  the  free-energy  content  (F),  or  simply 
the  free-energy  of  the  system,  is  defined  by  the  equation  F  =  A  -\-pv. 
Its  value  is  determined  by  the  state  of  the  system,  since  the  values  of 
A  and  of  p  v  are  so  determined ;  and  the  decrease  in  its  value  when 
any  change  in  the  state  of  the  system  takes  place  is  evidently  equal 
to  the  work  produced  when  the  change  takes  place  reversibly,  dimin- 
ished by  the  increase  of  the  pressure-volume  product;  that  is, 

^  "  P,  =  WR— (p.t;,— p,*,),   or  -AF=  WR-A(pv). 

The  difference  (F^ —  F2)  between  the  free-energy-content  of  a 
system  in  its  initial  state  and  that  in  its  final  state  will  be  called  the 
free-energy-decrease  ( — AF7)  attending  the  change  in  state,  irre- 
spective of  the  sign  of  its  numerical  value,  which  may  be  either  positive 
or  negative.  The  change  in  state  is  always  considered  to  take  place  at 
some  constant  temperature. 

Prob.  6.  a.  What  is  the  decrease  in  joules  in  the  work-content  and 
in  the  free-energy-content  of  one  formula-weight  of  water  when  it 
changes  from  liquid  water  at  100°  and  1  atm.  to  gaseous  water  at  100° 
and  1  atm.,  referring  to  Prob.  4,  Art.  109,  for  the  data  needed?  6.  What 
would  be  the  decrease  in  these  two  quantities  if  the  liquid  water  at  100° 
and  1  atm.  changed  to  gaseous  water  at  100°  and  1  atm.  by  a  process 
which  produces  no  work,  for  example,  by  introducing  the  liquid  water 
into  an  evacuated  vessel  having  a  volume  equal  to  that  of  the  saturated 
vapor? 

In  cases  where  different  parts  of  the  system  are  under  different 
pressures,  as  in  the  cell  of  Prob.  5,  the  decrease  in  free  energy  is 
defined  to  be  the  quantity  obtained  Vy  subtracting  from  the  work  pro- 
duced when  the  change  takes  place  reversibly  the  difference  between 
the  sum  of  the  p  v  values  for  all  the  parts  of  the  system  in  its  final 
state  and  the  sum  of  the  p  v  values  for  all  its  parts  in  its  initial  state. 
Therefore,  in  general : 
F,  —  F,  =  WR  —  2  (p2v,  —  PltO  ;v  or  —  AF  =  WR  —  2(Apv). 

The  principle  that  the  free-energy-decrease  attending  any  process 
is  determined  solely  by  the  change  in  state  of  the  system  is  the  funda- 


148  ELECTROCHEMISTRY 

mental  principle  on  which  is  based  the  treatment  presented  in  this  book 
of  the  subjects  of  Electrochemistry  and  Thermodynamic  Chemistry. 
The  first  step  in  any  application  of  the  principle  is  therefore  to  formu- 
late exactly  the  change  in  state  .of  the  system  in  terms  of  its  initial 
and  final  states;  and  the  next  step  is  to  formulate  an  expression  for 
the  free-energy-decrease  attending  it. 


FREE-ENERQY  CHANGES  149 

FREE-ENERGY    CHANGES   ATTENDING    CHANGES    IN    PRESSURE 
AND  CONCENTRATION 

126.  Free-Energy  Changes  attending  Changes  in  Volume  and 
Pressure.  —  When  a  system  which  is  under  an  external  pressure  equal 
to  its  own,  so  that  equilibrium  prevails,  undergoes  an  infinitesimal 
change  of  volume  dv  and  of  pressure  dp,  the  work  produced  dWR  is 
equal  to  pdv  (Art.  109).  The  free-energy-decrease  —  dF  attending 
such  a  change  in  state  at  a  constant  temperature  is,  however,  by  defini- 
tion equal  to  d\V  —  d(pv}.  In  virtue  of  the  mathematical  relation 


d(pv)  =  pdv  -f-  vdp,  the  free-energy-decrease  attending  an  infinites- 
imal change  in  volume  or  pressure  at  a  constant  temperature  is  there- 
fore given  by  the  expression  : 

—dF  =  —  vd. 


Prob.  7.  a.  Derive  an  expression  for  the  free-energy-decrease  (  — 
attending  the  change  in  state  of  N  mols  of  a  perfect  gas  when  at  the 
temperature  T  its  volume  and  pressure  change  from  vt  and  PI  to  va  and 
p2.  b.  Formulate  an  algebraic  expression  for  the  decrease  of  the  free 
energy  of  N  mols  of  hydrogen  when  at  18°  its  pressure  changes  from 
100  atm.  to  1  atm.  The  pressure-volume  relations  of  hydrogen  up  to  high 
pressures  are  expressed  by  the  equation  p  (v  —  Nb)  =  NRT,  in  which 
&  is  a  constant. 

127.  Free-Energy  Change  attending  the  Transfer  of  a  Substance 
from  a  Solution  of  One  Concentration  to  One  of  Another  Concentration. 

Prob.  8.  a.  Formulate  an  exact  expression  (not  involving  the  perfect- 
gas  law)  for  the  free-energy-decrease  attending  the  introduction  at  the 
temperature  T  of  m  grams  of  a  pure  substance  at  a  pressure  equal  to  its 
vapor-pressure  p0  into  an  infinite  quantity  of  a  solution  in  which  the 
substance  has  a  vapor-pressure  p.  Note  that  this  change  in  state  can  be 
brought  about  by  the  following  reversible  process  :  vaporize  the  m  grams 
of  the  substance  at  the  temperature  T  under  a  pressure  p0'»  change  the 
pressure  of  this  vapor  to  p  ;  and  at  this  pressure  condense  the  vapor  into 
the  solution.  I.  Derive  from  this  expression  an  equation  which  holds 
true  when  the  vapor  conforms  to  the  perfect-gas  law.  c.  Formulate  an 
expression  for  the  free-euergy-decrease  attending  the  transfer  of  N  mols 
of  a  substance  from  an  infinite  quantity  of  a  solution  in  which  its  vapor- 
pressure  is  p,  into  an  infinite  quantity  of  a  solution  in  which  its  vapor- 
pressure  is  />2. 

Prob.  9.  Show  how  the  equation  formulated  in  Prob.  8c  may  be 
modified  so  as  to  contain  the  mol-fractions  xl  and  x3  of  the  substance 
in  the  two  solutions,  «,  when  the  vapor-pressures  pa  and  P2  conform  to 
Raoult's  law:  b,  when  these  vapor-pressures  conform  to  Henry's  law. 


150  ELECTROCHEMISTRY 

c.  State  in  the  case  of  a  solution  consisting  of  two  substances  A  and  B 
the  conditions  of  composition  under  which  Raoult's  law  and  under  which 
Henry's  law  (and  therefore  under  which  the  corresponding  free-energy 
expressions)  hold  true  approximately,  d.  Show  as  in  Art.  27  that  when 
the  mol-fractions  x^  and  xz  are  small,  their  ratio  may  be  replaced  by  the 
ratio  of  the  concentrations  cx  and  c2,  which  denote  the  number  of  mols 
of  the  solute  per  unit-volume  of  the  solvent. 

In  the  preceding  problems  the  following  expressions  have  been 
derived  for  the  free-energy-decrease  which  attends  the  transfer  at  the 
temperature  T  of  that  quantity  of  a  substance  which  is  N  mols  in 
the  state  of  a  perfect  gas  from  an  infinite  quantity  of  a  solution 
in  which  its  vapor-pressure  is  pv  its  mol-fr action  xlt  and  its  concen- 
tration Cj,  into  an  infinite  quantity  of  another  solution  in  which  its 
vapor-pressure  is  p2,  its  mol-fraction  xv  and  its  concentration  ca: 

—  £**  =  #£  T  log  ?L(1);    -&F  =  NRT\og^(2); 

P2  #2 

—  &F  =  NRT  log^i   (3). 
cz 

Equation  (1)  holds  true  whatever  be  the  mol-fractions  or  concentra- 
tions; but  it  involves  the  assumption  that  the  vapor  conforms  to  the 
perfect-gas  law.  Equation  (2)  holds  true  when,  in  conformity  either 
with  Raoult's  law  or  with  Henry's  law,  the  vapor-pressures  are  pro- 
portional to  the  mol-fractions.  Equation  (3)  holds  true  when,  in 
conformity  with  Henry's  law,  the  vapor-pressures  are  proportional 
to  the  mol-fractions  and  when  the  latter  are  small  enough  to  be  sub- 
stantially proportional  to  the  concentrations;  it  is  the  expression 
commonly  employed  when  a  solute  is  transferred  from  one  dilute 
solution  to  another. 

Since  Henry's  law  applies  to  the  concentrations  of  a  definite 
molecular  species,  it  is  evident  that  equation  (3)  must  be  applied 
separately  to  the  transfer  of  each  molecular  species,  not  to  the  transfer 
of  the  substance  as  a  whole,  if  it  exists  in  the  solution  as  two  or  more 
different  molecular  species,  as  is  the  case  with  H+C1"  or  any  other 
partially  ionized  substance  in  aqueous  solution,  or  with  ac^ic  acid  in 
benzene  solution  in  which  it  exists  as  C2H4Oa  and  (CaH4O2)2.  More- 
over, although  equation  (3)  was  derived  by  the  consideration  of  ft 
process  involving  the  vaporization  of  the  solute  and  with  the  aid  of 
the  assumption  that  the  pressure  of  the  vapor  was  small  enough 


FREE-ENERGY  CHANGES  151 

to  conform  to  the  perfect-gas  law,  yet  that  equation  relates  only  to 
the  transfer  of  the  substance  from  one  solution  to  another;  and  it 
would  be  remarkable  if  its  validity  depended  on  whether  the  substance 
were  volatile  or  on  how  large  its  vapor-pressure  might  be.  In  fact,  it 
can  be  shown,  by  deriving  equation  (3)  through  a  consideration  of  an 
osmotic  process  of  transferring  the  substance  from  one  solution  to  the 
other,  that  the  equation  is  exact,  provided  only  that  the  substance 
behaves  as  a  perfect  solute,  as  shown  by  its  conformity  to  the  osmotic- 
pressure  equation  P  =  c  R  T  considered  in  Art.  38. 

Prob. 10.  a.  Calculate  the  free-energy-decrease  attending  the  transfer 
at  20°  of  1NH,  from  an  infinite  quantity  of  a  solution  of  the  composition 
1XH38^H2O  in  which  its  vapor-pressure  is  80  mm.  into  an  infinite  quan- 
tity of  a  solution  of  the  composition  1NHR21H2O  in  which  its  vapor- 
pressure  is  27  mm.  ft.  Calculate  the  free-energy-decrease  attending  the 
dissolving  «t  20°  of  1NH,  at  1  atm.  in  an  infinite  quantity  of  a  solution 
of  the  composition  1NH321H2O. 

Note. — Throughout  this  chapter  free-energy  values  are  to  be  expressed 
in  joules.  For  the  gas-constant  R  its  value  8.32  in  jou'es  per  degree 
(derived  in  Prob.  24,  Art.  22)  must  therefore  be  used. 

Prob.  11.  Calculate  the  free-energy-decrease  attending  the  transfer 
at  20°  of  INHs  from  an  infinite  quantity  of  0.1  formal  NH4OH  solution 
into  an  infinite  quantity  of  0.001  formal  NH4OH  solution.  The  ammonium 
hydroxide  is  1.3%  ionized  in  the  0.1  formal  and  12.5%  ionized  in  the 
0.001  formal  solution,  and  the  remaining  ammonium  hydroxide  is  m 
percent  dissociated  into  NH3  and  H2O  in  each  solution. 

Prob.  12.  The  transfer  at  the  temperature  T  of  one  formula-weight  of 
NaCl  from  an  infinite  quantity  of  a  dilute  solution  in  which  its  formal 
concentration  is  c^  and  its  ionization  is  yt  into  an  infinite  quantity  of 
another  dilute  solution  in  which  its  formal  concentration  is  c2  and  its 
ionization  is  y^  can  be  brought  about  by  so  transferring  either  1  mol  Na* 
and  1  mol  Cl-  or  1  mol  unionized  NaCl.  a.  Formulate  an  expression  for 
the  free-energy-decrease  attending  each  of  these  processes.  b.  Show  that 
one  of  these  expressions  can  be  derived  from  the  other  with  the  aid  of 
the  mass-action  equation  for  the  ionization  of  the  salt.  c.  Calculate  the 
free-energy-decrease  by  both  expressions  for  the  case  that  the  tempera- 
ture is  18°,  the  concentrations  are  0.0334  and  0.00167  formal,  and  the 
ionizations  are  0.900  and  0.970. 

The  large  divergence  between  the  values  of  the  free-energy-decrease 
calculated  in  Prob.  12  by  the  two  expressions  corresponds  to  the  devia- 
tions (considered  in  Art.  86)  from  the  requirements  of  the  mass-action 
law  exhibited  by  largely  ionized  substances.  It  shows  that  either  the 


152  ELECTROCHEMISTRY 

ions  or  the  unionized  substance  or  both  deviate  considerably  from 
the  laws  of  perfect  solutions  (thus  from  the  requirements  of  the 
osmotic-pressure  equation  P  =  cRT).  It  will  be  seen  later  (in 
Prob.  23,  Art.  131)  that  the  true  value  of  the  free-energy-decrease 
attending  the  transfer  of  the  sodium  chloride  can  be  directly  derived 
from  the  electromotive  force  of  a  cell  containing  the  two  sodium 
chloride  solutions.  This  true  free-energy-decrease  is  larger  by  26  per- 
cent of  its  value  than  that  calculated  by  considering  the  transfer  of 
the  unionized  substance;  and  it  is  smaller  by  3.0  percent  of  its  value 
than  that  calculated  by  considering  the  transfer  of  the  ions.  This 
result  is  representative  of  the  behavior  of  largely  ionized  uniunivalent 
substances  in  general.  It  shows  that  the  unionized  substance  deviates 
very  greatly  from  the  laws  of  perfect  solutions,  and  that  a  great  error 
would  be  made  if  it  were  assumed  to  conform  to  those  laws  in  mass- 
action  or  in  free-energy  calculations.  The  comparison  also  shows  that 
the  ions  deviate  considerably  from  the  behavior  of  perfect  solutes ;  the 
deviation  up  to  0.1  formal  is,  however,  not  so  great  as  to  cause  a  very 
serious  error  in  approximate  calculations.  Therefore,  throughout  this 
book,  in  the  case  of  largely  ionized  substances,  the  ions,  rather  than 
the  unionized  substance,  are  always  considered;  and  the  deviation 
of  the  activity  of  the  ions  of  uniunivalent  substances  from  that  of 
perfect  solutes  up  to  a  concentration  of  0.1  formal  is  disregarded. 

In  the  case  of  unibivalent  salts,  such  as  zinc  chloride  or  sodium 
sulphate,  for  which  the  ionization  values  are  uncertain  owing  to  the 
probable  presence  of  intermediate  ions  (referred  to  Art.  58),  the  free- 
energy-decrease  attending  their  transfer  from  one  solution  to  another 
can  be  calculated  even  approximately  only  when  the  concentrations 
are  so  small  (say  less  than  0.01  formal)  that  no  great  error  is  made  by 
regarding  the  ionization  as  complete. 


• 

VHANGES  IN  STATE  IN  CELLS  153 

CHANGES  IN   STATE  AND   PRODUCTION  OF  WORK  IN   VOLTAIC   CELLS 

128.  Changes  in  State  in  Voltaic  Cells. — A  change  in  state  can  be 
made  to  yield  electrical  energy  when  it  can  be  brought  about  by  a 
process  of  reduction  occurring  in  one  place  and  a  process  of  oxidation 
occurring  in  another  place;  for  it  is  an  inherent  characteristic  of 
reduction  processes  that  they  are  attended  by  a  liberation  of  positive 
electricity  or  by  an  absorption  of  negative  electricity,  and  of  oxidation 
processes  that  they  are  attended  by  the  opposite  electrical  effects. 
Thus  the  reduction  of  copper-ion  to  metallic  copper  may  be  represented 
by  the  equation  Cu++  =  Cu  +  2©  ;  and  the  oxidation  of  metallic  zinc 
to  zinc- ion,  by  the  equation  Zn  -f-  2  0  =  Zn"1"1",  where  the  symbol  © 
denotes  one  faraday  of  positive  electricity.  If  these  two  processes 
occur  at  the  same  place,  as  is  the  case  when  metallic  zinc  is  placed  in 
a  copper-ion  solution,  no  electrical  effect  is  observed.  If,  however,  the 
zinc  is  placed  in  a  zinc-ion  solution  and  the  copper  in  a  copper-ion 
solution,  and  if  the  two  solutions  are  placed  in  contact,  the  reduction- 
process  tends  to  occur  at  one  place  and  to  liberate  positive  electricity 
there,  and  the  oxidation-process  tends  to  occur  at  another  place  and 
to  absorb  positive  electricity  there,  thereby  producing  a  difference  of 
potential  or  an  electromotive  force  between  the  two  places ;  and  if  they 
are  now  connected  by  a  metallic  conductor,  a  current  of  electricity 
will  flow  through  -it,  and  this  current  can  be  made  to  produce  work, 
for  example,  by  passing  it  through  an  electric  motor.  Such  an  arrange- 
ment as  that  here  described,  in  which  an  electric  current  is  produced 
by  causing  an  oxidation-process  and  a  reduction-process  to  occur  at 
two  different  places,  is  known  as  a  voltaic  cell. 

It  has  been  shown  in  Art.  125  that  the  quantity  of  work  that  can 
be  produced  by  any  process  is  determined  solely  by  the  change  in  state 
of  the  system.  Hence  the  maximum  quantity  of  work  producible  by 
the  action  of  a  voltaic  cell  is  fully  determined  by  the  initial  and  final 
states  of  the  substances  of  which  it  consists;  and  in  any  case  under 
consideration  these  states,  or  the  corresponding  change  in  state,  must 
be  exactly  specified.  Thus,  there  must  be  stated,  in  addition  to  the 
temperature,  not  merely  the  chemical  reaction  that  takes  place  in 
the  cell,  but  also  the  conditions  of  pressure  and  concentration  under 
which  the  substances  involved  in  it  are  produced  and  destroyed  and 


154  ELECTROCHEMISTRY 

any  transfers  of  substances  from  solutions  of  one  composition  to  those 
of  another  composition. 

In  order  that  there  may  not  be  a  finite  change  in  the  concentrations 
of  the  solutions,  and  therefore  in  the  electromotive  force  of  the  cell, 
during  the  occurrence  of  the-  change  in  state,  it  will  always  be  assumed 
that  the  solutions  are  present  in  infinite  quantity,  so  that  when  a 
finite  quantity  of  any  substance  is  introduced  into  or  withdrawn  from 
one  of  the  solutions  of  the  cell  there  is  only  an  infinitesimal  change 
in  its  concentration. 

The  character  of  the  cell  under  consideration  will  be  shown  by 
writing  the  symbols  of  the  pure  substances  and  solutions  in  the  order 
in  which  they  are  actually  in  contact  with  one  another,  commas  being 
inserted  to  indicate  the  junctions  at  which,  as  will  be  explained  later, 
an  electromotive  force  is  produced.  The  conventions  described  in 
Art.  113  will  be  employed  to  indicate  the  state  of  aggregation  of  sub- 
stances and  the  composition  of  solutions.  Thus  a  cell  consisting  in 
series  of  metallic  zinc,  of  a  zinc  chloride  solution  of  the  composition 
ZnCl2100H2O,  of  solid  mercurous  chloride,  and  of  metallic  mercury, 
at  a  pressure  of  one  atmosphere  (as  is  always  understood  unless  some 
other  pressure  is  specified),  will  be  represented  by  the  expression: 

Zn,  Zn012100H2O,  Hg2Cl2-f-Hg. 

Similarly,  a  cell  whose  electrodes  are  an  inert  metal  M  in  contact  with 
hydrogen  gas  at  0.1  atmosphere  and  the  same  metal  in  contact  with 
chlorine  gas  at  0.05  atmosphere,  and  whose  electrolyte  is  a  0.1  formal 
HC1  solution,  will  be  represented  by  the  expression: 

M  +  fl,(l  atm.),  HC1(0.1  1),  CZa(0.05  atm.)  +  M. 
So  also  a  lead  storage-cell  whose  electrodes  are  lead  and  an  inert  metal 
coated  with  lead  dioxide,  and  whose  electrolyte  is  a  sulphuric  acid 
solution,  say  of  the  composition  H2SO410H2O,  saturated  with  lead 
sulphate,  will  be  represented  by  the  expression: 

Pb  4-  PbS04,  H2SO410H2O,  PbS04  +  Pb02  +  M. 
The  change  in  state  taking  place  in  a  cell  may  he  expressed  by  an 
equation  whose  left-hand  member  represents  the  initial  state,  and 
whose  right-hand  member  represents  the  final  state,  of  the  substances 
involved  in  the  change.  Thus  the  changes  of  state  occurring  when  at 
25°  two  faradays  pass  from  left  to  right  (as  is  always  understood 


CHANGES  IN  STATE  IN  CELLS  155 

unless  the  opposite  is  specified)  through  the  first  two  cells  just  formu- 
lated are  shown  by  the  equations : 

Zn  +  Hg2CL=:  2Hg  +  ZnCl2(in  ZnOl.lOOH.O)  at  25° 
H2(l  atm.)  -f  C72(0.05  atm.)  =  2HCl(at  0.1  f.)  at  25°. 

In  determining  the  change  in  state  in  the  cell  it  is  often  well  to 
consider  separately  the  reactions  that  take  place  at  the  two  electrodes 
(see  Art.  42)  and  the  ion-transferences  that  occur  in  the  solutions. 
The  electrode-reactions  take  place  in  accordance  with  Faraday's  law 
(Art.  43),  and  the  transference-effects  in  accordance  with  the  principles 
of  transference  (Arts.  45-49). 

1'rob.  13.  a.  State  the  changes  at  each  electrode  and  the  ion-transfer- 
ences that  attend  the  passage  of  two  faradays  through  the  first  cell 
formulated  above;  and  show  that  the  net  result  of  these  changes  is  the 
change  in  state  expressed  by  the  first  of  the  preceding  equations,  ft.  Do 
the  same  for  the  second  cell  formulated  above. 

Prol).  llf.  a.  State  the  electrode  changes  and  ion-transferences  that 
occur  when  two  faradays  pass  at  20°  through  the  lead  storage-cell  formu- 
lated above.  I).  Express  the  resultant  change  of  state  that  takes  place 
in  the  cell  by  an  equation. 

When,  as  in  these  cells,  a  gaseous  or  a  solid  non-metallic  substance 
is  in  contact  with  a  metal  electrode,  the  adjoining  solution  is  under- 
stood to  be  saturated  with  that  substance.  In  such  cells  there  are  there- 
fore in  reality  two  different  solutions,  even  though  for  the  sake  of 
brevity  only  one  may  be  written  in  formulating  the  cell;  thus  these 
cells  ought  strictly  to  be  written : 

Zn,   ZnCL100H20,   ZnCl2100H2O -f  Hg2Cl2(s.  f.),*    Hg2Cl,4-Hg. 

M  +  #2(1  atm.),   H01(0.1f.)  -f  H2 (0.00076  f.),   HC1(0.1  f.)  + 

Cla(0.0044f.),  CZ,(0.05  atm.)  +  M. 
Pb  +  Pt>S04,  H2SO410H2O+PbSO4(s.f.),  H2SO410H2O+PbS04.(s.f.) 

+  Pb(SO4)2  (10-8  f.),   PbS04  +  Pb02  +  M. 

The  recognition  of  the  fact  that  there  are  in  such  cells  two  solutions 
of  slightly  different  composition  is  of  great  importance  in  considering 
the  mechanism  of  voltaic  action  and  in  evaluating  the  separate  poten- 
tials discussed  in  Arts.  133-137. 

When,  as  in  the  three  cells  above  considered,  the  cell  contains  two 
solutions  of  substantially  the  same  composition  (in  the  respects  that 

*The  symbol  s.  f.  represents  the  solubility  (in  formula-weights  per  1000  g. 
of  water)  of  Hg2Cl2  in  a  solution  of  composition  ZnCla100HaO. 


156  ELECTROCHEMISTRY 

in  both  solutions  the  ions  present  at  considerable  concentrations  are 
the  same,  that  the  concentrations  of  these  ions  are  substantially  the 
same,  and  that  the  solvent-medium  as  a  whole  is  substantially  the 
same) ,  the  ion- transferences  need  not  be  considered ;  for,  though  there 
is  transference  from  the  solution  around  one  electrode  to  that  around 
the  other  electrode,  there  is  not  an  appreciable  free-energy  change, 
since  the  two  solutions  have  substantially  the  same  composition. 

Pro 6.  15.  Specify  the  electrode-changes  and  ion- transferences  and 
the  resultant  change  in  state  taking  place  when  one  faraday  passes 
through  each  of  the  following  cells: 

a.    #2(1  atm.),  HC1  (0.01  f.),  HC1(0.1  f.),  #,(1  atm.). 

&.    CZ2(1  atm.),  HC1(0.01  f.),  HC1(0.1  1),  Cl2(l  atm.). 
Note  that  the  transference-effects  in  these  cells  are  similar  to  those 
described  in  Art.  45,  the  left-hand  solution  being  regarded  as  the  anode- 
portion  and  the  right-hand  solution  as  the  cathode-portion. 

129.  Production  of  Work  in  Voltaic  Cells. — According  to  a  funda- 
mental principle  of  the  science  of  electricity,  when  a  quantity  of 
positive  electricity  Q  flows  between  two  places,  such  as  the  electrodes 
of  a  voltaic  cell,  between  which  there  is  a  potential-difference  or  electro- 
motive force  E,  a  quantity  of  work  equal  to  the  product  EQ  can  be 
produced. 

In  order  that  the  work  produced  by  a  voltaic  cell  may  be  the 
maximum  which  can  be  obtained  from  the  change  in  state  taking  place 
in  it,  the  electromotive  force  at  the  electrodes  must  be  such  that, 
when  it  is  increased  by  an  infinitesimal  amount,  the  change  in  state 
under  consideration  takes  place  in  the  opposite  direction.  It  is  this 
value  of  the  electromotive  force  which  is  considered  throughout  the 
following  pages,  and  which  is  called  the  electromotive  force  (E)  of 
the  cell.  In  an  experimental  determination  of  this  electromotive  force 
the  assurance  must  be  obtained,  by  making  measurements  under  vari- 
ous conditions,  that  the  measured  electromotive  force  really  corre- 
sponds to  the  change  of  state  under  consideration,  and  not  to  some 
incidental  process  taking  place  at  the  electrodes. 

Now,  according  to  Faraday's  law,  the  quantity  of  electricity  flow- 
ing through  a  voltaic  cell  is  strictly  proportional  to  the  number  of 
equivalents  N  that  are  involved  in  the  chemical  change  at  each  elec- 
trode ;  that  is,  Q  =  N  F,  where  F  represents  the  quantity  of  electricity 
(96,500  coulombs)  that  passes  when  a  reaction  involving  one  equiva- 
lent of  each  of  the  reacting  substances  takes  place  at  each  electrode. 


PRODUCTION  OF  WORK  IN  CELLS  157 

The  electrical  work  that  can  be  produced  by  a  change  in  state 
taking  place  in  a  voltaic  cell  and  involving  the  passage  of  N  faradays 
of  electricity  is  therefore  equal  to  E  N  F. 

The  numerical  value  of  the  electromotive  force  E  of  the  cell  will 
in  this  book  be  given  a  positive  sign  when  the  cell  tends  to  produce 
a  current  of  positive  electricity  through  the  cell  in  the  direction  in 
which  it  is  written,  and  a  negative  sign  when  the  cell  tends  to  pro- 
duce such  a  current  in  the  opposite  direction.  The  symbol  N  will 
denote  the  number  of  faradays  of  positive  electricity  that  are  con- 
sidered to  pass  through  the  cell  from  left  to  right,  its  numerical  value 
being  given  a  negative  sign  when  positive  electricity  is  considered 
to  pass  in  the  opposite  direction.  The  value  of  W  calculated  by  the 
equation  will  then  have,  as  usual,  a  positive  sign  when  the  cell  pro- 
duces electrical  work,  and  a  negative  sign  when  external  work  is 
expended  upon  the  cell.  The  work  will  be  in  joules  when  the  electro- 
motive force  is  in  volts  and  the  quantity  of  electricity  in  coulombs. 

Prol).  16.  The  electromotive  force  at  15°  of  the  Daniell  cell  Zn, 
ZnS047H20  -f  ZnSO4w'H2O,  CuSO4n"H2O  +  CuS045H20,  Cu  is  1.093  volts. 
What  must  be  the  values  of  NF  in  coulombs  and  of  W  in  joules  in  order 
to  precipitate  1  at.  wt.  of  zinc ;  and  what  does  the  sign  of  each  of  these 
quantities  signify? 

When  a  voltaic  cell  acts  reversibly  there  is  ordinarily  produced, 
in  addition  to  the  electrical  work,  a  quantity  of  mechanical  work 
corresponding  to  the  changes  in  volume  of  the  different  parts  of  the 
cell  taking  place  under  their  respective  pressures.  The  total  work  W  B 
is  therefore  E  N  F  -f-  S(joAv).  Since  by  Art.  125  the  free-energy- 
decrease  is  equal  to  W^  — £(Ap  t>),  it  is  equal  simply  to  the  elec- 
trical work  that  can  be  produced.  That  is,  for  any  change  in  state 
taking  place  in  a  voltaic  cell  under  constant  pressures : 

—  AF  =  ENF. 

Prol).  17.  Calculate  exact  values  of  the  electrical  work  and  the 
mechanical  work  that  are  produced  when  1  faraday  passes  under  re- 
versible conditions  through  the  cell  -Hj(l  atm.),  HC1(0.1  f.,  1  atui.),  Cl, 
(0.05  atm.)  at  25°.  The  electromotive  force  of  this  cell  is  1.451  volts. 
The  increase  in  the  volume  of  an  infinite  quantity  of  0.1  f.  HC1  solution 
caused  by  introducing  1HC1  into  it  is  18.7  ccm.  What  is  the  correspond- 
ing free-energy-decrease? 


158  ELECTROCHEMISTRY 

ELECTROMOTIVE   FORCE  OP  VOLTAIC  CELLS   IN   RELATION 
TO  CONCENTRATION 

130.  Change  of  the  Electromotive  Force  of  Voltaic  Cells  with  the 
Concentration  of  the  Solutions. — The  considerations  of  the  preceding 
articles  make  it  possible  to  calculate  the  change  that  is  produced  in 
the  electromotive  force  of  a  cell  by  varying  the  concentration  of  the 
solutions  contained  in  it.    Thus  the  difference  between  the  electro- 
motive force  of  the  cell  H2(l  atm.),    HC1(0.01  f.),    Cl2(l  atm.)  and 
that  of  the  cell  H2(l  atm.),   HC1(0.1  f.),  Cl2(l  atm.)  can  be  derived 
by  considering  that  one  faraday  passes  through  these  two  cells  arranged 
in  series  in  opposition  to  each  other,  by  noting  what  the  resultant 
change  in  state  is,  and  by  equating  the  two  expressions  for  the  attend- 
ant free-energy-decrease  derived  in  preceding  articles. 

Pro b.  18.  Calculate  the  difference  between  the  electromotive  forces 
at  18°  of  the  two  cells  named  in  the  preceding  text.  The  ionizations  of 
the  acid  in  the  two  solutions  are  0.972  and  0.925. 

Prob.  19.  Calculate  the  difference  between  the  electromotive  forces 
at  25°  of  the  cells  J/2(l  atm.),  HC19H2O,  Cl2(l  atm.)  and  #2(1  atm.), 
HC116.7H2O,  C12(1  atm.).  The  vapor-pressures  of  HC1  in  the  two  solu- 
tions at  25°  are  0.0550  and  0.0044  mm. 

Prob.  20.  State  what  data  are  needed  in  order  to  calculate  the 
electromotive  force  Et  at  18°  of  the  cell  If2  (1  atm.),  H,SO4(10f.),  Oa(latm.) 
from  that  E2  of  the  cell  H2(l  atm.),  H2SO4(0.01  f.),  O,(l  atm.) ;  and 
formulate  an  expression  by  which  the  calculation  could  be  made. 

Prob.  21.  The  cell  Ag-f  AgCl,  HCl(0.1f.),  CZ2(latm.)  has  at  25°  an 
electromotive  force  of  1.142  volts.  How  much  would  its  electromotive 
force  be  changed  by  substituting  HC1(0.01  f.)  for  the  HC1(0.1  f.)? 

131.  The   Electromotive   Force   of   Concentration-Cells. — A   cell 
which  consists  of  two  identical  electrodes  and  of  two  solutions  con- 
taining the  same  substance  at  two  different  concentrations  is  cal'ed 
a  concentration-cell    The  cell  Zn,  ZnCl2(0.01  f.),  ZnCi(0.001  f.),  Zn 
is  an  example  of  a  concentration-cell;  so  also  are  the  cells  formulated 
in  Prob.  15. 

Prob.  22.  a.  Derive  an  algebraic  expression  for  the  electromotive 
force  of  the  cell  #2(1  atm.),  HCl(c'  f.),  HCl(c"  f.),  #2(1  atm.),  where 
the  concentrations  c'  and  c"  are  small,  by  considering  (as  in  Prob.  15)  the 
change  in  state  that  occurs  in  the  cell  when  N  faradays  pass  through  it 
and  by  considering  the  two  expressions  for  the  attendant  free-energy- 
decrease  derived  in  preceding  articles.  b.  Calculate  the  electromotive 
force  of  the  cell  at  18°  when  the  concentrations  c7  and  c"  are  0.01  and 
0.1  formal.  For  the  ion-conductances  see  Art.  56. 


CONCENTRATION-CELLS  159 

Profi.  23.  a,  b.  Answer  the  same  questions  as  in  Prob.  22  for  the  cell 
atm.),  HCl(c'  f.),  HCl(c"  f.),  Clz(l  atm.). 

Prob.  2.'f.  Formulate  a  numerical  expression  by  which  the  electro- 
motive force  of  the  cell  Zn,  ZnCl2(0.01  f.),  ZnCl, (0.001  f.),  Zn  at  18°  can 
be  calculated.  Consider  the  ionization  of  the  salt  in  both  solutions  to  be 
complete.  (See  the  last  paragraph  of  Art.  127.) 

Prob.  25.  Calculate  the  electromotive  force  at  18°  of  the  cell  Ag  -{- 
AgCl,  NaCl (0.0334  f.),  NaCl (0.00167  f.),  AgCl  +  Ag,  using  the  data  of 
Prob.  12c.  (The  electromotive  force  of  this  cell  has  been  experimentally 
determined  and  found  to  be  0.05614  volt.) 

Another  type  of  concentration-cell  is  that  in  which  there  is  a  single 
aqueous  solution  in  contact  with  electrodes  consisting  of  some  metal 
dissolved  at  two  different  concentrations  in  mercury.  For  example, 
the  cell  ZnlOOHg,  Zn012100H2O,  Zn200Hg  is  of  this  type.  Since 
most  of  the  metals  dissolved  at  small  concentrations  in  mercury  have 
been  shown  by  vapor-pressure  and  freezing-point  measurements  to  be 
nearly  perfect  solutes  having  monatomic  molecules,  the  electromotive 
force  of  cells  having  such  solutions  as  electrodes  can  be  calculated 
with  the  aid  of  the  preceding  considerations. 

Another  similar  type  of  concentration-cell  is  that  in  which  the  two 
electrodes  consist  of  some  inert  metal  surrounded  by  the  same  gas  at 
two  different  pressures,  as  is  the  case  in  the  hydrogen  cell  of  Prob.  5 
and  in  the  oxygen  cell  M  +  02(1  atm.),  KOH(1  f.),  02(0.1  atm.)  +  M. 

Pro b.  26.  Calculate  the  electromotive  force  at  18°  of  the  zinc-mercury 
cell  formulated  in  the  preceding  text. 

Prob.  27.  Calculate  the  electromotive  force  at  25°  of  the  oxygen  cell 
formulated  in  the  preceding  text. 


160  ELECTROCHEMISTRY 

. 

ELECTROMOTIVE  FORCE  OF  CELLS  INVOLVING  CHEMICAL   CHANGES. 
ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS 

132.  Chemical  Changes  in  Voltaic  Cells. — In  the  cells  thus  far 
considered  the  change  of  state  consists  only  in  a  transfer  of  one  or 
more  of  the  substances  from  one  pressure  or  concentration  to  another. 
In  most  voltaic  cells,  however,  a  chemical  change  takes  place.    Thus 
in  the  Daniell  cell  the  reaction  Zn  -f  CuS04  =  Cu  -}-  ZnS04  occurs. 
Other  examples  of  cells  in  which  similar  chemical  changes  take  place 
are  given  in  Art.  128. 

A  chemical  change  somewhat  different  in  character  from  those 
just  considered  takes  place  in  cells  whose  half -cells  consist,  not  of  solid 
or  gaseous  elementary  substances  in  contact  with  solutions  of  their 
ions,  but  of  an  inert  metal  electrode  in  contact  with  two  solutes  in 
different  stages  of  oxidation.  Thus  the  electrode-reactions  in  the  cell 
M,  Fe^Cl-.CcJ+Fe^+Ol-^c,),  Cr(e.)  +  Cla(c4),  M,  attending  the 
passage  of  one  faraday  are  Fe++  +  0  =  Fe+++  and  iCl,  =  Cl-+e, 
and  the  whole  reaction  is  Fe++Cl-2  -f  iCl2  =  Fe+++Cl-3.  Such  cells  do 
not,  however,  require  special  treatment;  for  the  principles  applicable 
to  other  cells  can  be  readily  extended  to  them,  as  will  be  seen  in  the 
following  articles. 

Evidently,  if  the  free-energy-decrease  attending  the  chemical 
change  in  any  cell  could  be  determined  by  any  independent  method, 
the  electromotive  force  of  the  cell  could  be  found  with  the  aid  of  the 
equation  — A^  — ENF.  There  is,  in  fact,  such  a  method  by  which 
the  free-energy-decrease  attending  a  chemical  change  can  be  calcu- 
lated when  the  pressures  or  concentrations  involved  are  small.  This 
method  will  be  considered  in  the  following  chapter. 

In  this  chapter  will  be  presented  only  the  electromotive-force  side 
of  the  problem.  There  remains  therefore  to  be  considered  only  the 
partial  electromotive  forces  at  the  different  junctions  within  the  cell. 
The  consideration  of  these  is  important  in  two  respects :  first,  it  shows 
more  clearly  the  separate  factors  which  determine  electromotive  force : 
and  secondly,  it  enables  the  electromotive  force  of  a  great  number  of 
cells  to  be  calculated  from  a  relatively  small  number  of  experimentally 
determined  constants. 

133.  The  Electrode-Potentials  of  Voltaic  Cells.— The  electromotive 
force  produced  by  a  voltaic  cell  is  the  sum  of  the  electromotive  forces 


ELECTRODE-POTENTIALS  161 

produced  at  the  junctions  between  the  electrodes  and  the  solutions 
and  of  those  produced  at  the  junctions  between  the  different  solutions 
that  may  be  present  in  the  cell.  Thus  the  electromotive  force  of  the 
Daniell  cell  is  the  algebraic  sum  of  the  electromotive  force  from 
the  zinc  to  th€  zinc-sulphate  solution,  that  from  the  zinc-sulphate 
solution  to  the  copper-sulphate  solution,  and  that  from  the  copper- 
sulphate  solution  to  the  copper.  The  electromotive  forces  at  the  elec- 
trodes are  commonly  called  electrode-potentials;  and  those  at  the 
junctions  of  the  solutions,  liquid-potentials. 

Although  attempts  have  been  made  to  determine  by  certain  experi- 
mental methods  the  absolute  values  of  electrode-potentials,  yet  the 
measurements  made  by  these  methods  are  not  nearly  so  accurate  nor 
reliable  as  those  of  the  electromotive  force  of  ordinary  cells.  It  is 
therefore  customary  to  adopt  as  the  value  of  the  electrode-potential 
of  any  half -cell  (such  as  Zn,  ZnCl,100H2O)  the  electromotive  force  of 
the  whole  cell  which  consists  of  the  half-cell  under  consideration 
combined  with  a  standard  half-cell.  The  electrode-potential  of  the 
standard  half-cell  is  thereby  arbitrarily  assumed  to  be  zero.  Three 
different  half -cells  are  in  use  as  such  standards  of  reference;  namely, 
the  half-cell  Hg,  Hg2Cl_,  +  K01(l  n.),  the  half-cell  H2(l  atm.),  H2SO4 
(2  n.),  and  the  half -cell  7J2(1  atm.),  H+(l  m.).  The  first  of  these  cells, 
which  is  commonly  called  the  calomel-electrode,  has  certain  experi- 
mental advantages.  The  last  of  these  cells,  which  will  be  here  called 
the  molal  hydrogen-electrode,  has  the  simplest  theoretical  significance. 
The  exact  difference  between  the  electrode-potentials  of  these  standard 
half-cells  will  be  shown  later. 

The  electrode-potential  of  any  half-cell  is  therefore  equal  to  the 
electromotive  force  of  the  whole  cell  formed  by  combining  it  with 
one  of  these  standard  half-cells.  Thus  the  electrode-potential  of  the 
half -cell  Zn,  ZnCl2100H2O  referred  to  the  calomel-electrode  is  equal  to 
the  electromotive  force  of  the  whole  cell  Zn,  ZnO^lOOHO  1 1  KC1(1  n.) 
-f-  Hg^Cl,,  Hg.  Similarly  the  electrode-potential  of  the  half-cell  C12 
(latm.),  HCl(0.1f.)  referred  to  the  molal  hydrogen-electrode  is 
equal  to  the  electromotive  force  of  the  cell  C12(1  atm.),  HCl(0.1f.)  | , 
H*(lm.),  #2 (latm.).  It  is,  however,  further  understood  that  in 
evaluating  this  electromotive  force  the  liquid-potential  has  been  sub- 
tracted from  the  measured  electromotive  force  of  the  whole,  cell — a 


162  ELECTROCHEMISTRY 

fact  which  is  indicated  in  the  symbolic  representation  of  the  cell  by 
inserting  parallel  lines,  instead  of  a  comma,  at  the  liquid  junction. 

It  will  be  noted  that  the  electrode-potential  has  a  positive  sign 
when  positive  electricity  tends  to  flow  from  the  electrode  to  the 
solution,  and  a  negative  sign  when  it  tends  to  flow  in  the  opposite 
direction.  In  the  case  of  liquid-potentials  the  same  convention  as 
to  sign  is  adopted  as  in  the  case  of  whole  cells;  namely,  the  liquid- 
potential  is  given  a  positive  sign  when  positive  electricity  tends  to 
flow  from  the  solution  whose  symbol  is  written  on  the  left-hand  side 
to  the  solution  whose  symbol  is  written  on  the  right-hand  side;  and 
it  is  given  a  negative  sign  in  the  reverse  case.  (In  using  data  from 
outside  sources,  it  is  to  be  noted  that  foreign  electrochemists  employ 
the  opposite  convention  as  to  the  sign  of  electrode-potentials.) 

Prob.  28.    Measurements  of  the  electromotive  forces  at  25°  of  the  cells 
#2(1  atm.),  HC1(0.1  n.),  KC1(0.1  n.),  KC1(1  n.),  H&Clj-fHg, 
C'Z,(1  atm.),  HCKO.l  n.),  KC1(0.1  n.),  KC1(1  n.),  Hg^  +  Hg, 
have  given  the  values  0.3740  and  —  1.114  volts,  respectively.   The  liquid- 
potential  of  KC1(0.1  n.),  HC1(0.1  n.)  has  been  found  to  be  —  0.0283  volt; 
and  that  of  KCHO.l  n.),  KC1(1  n.)  is  probably  about  0.001  volt.   a.  Find 
the  values  of  the  electrode-potentials  that  can  be  derived  from  these  data. 
I).  Calculate  the  electromotive  force  of  the  cell  H2(l  atm.),  HC1(0.1  n.), 


134.  Change  of  Electrode-Potentials  with  the  Ion-Concentrations. 
Concept  of  Specific  Electrode-Potentials.  —  Just  as  the  total  electro- 
motive force  of  a  cell  is  determined  solely  by  the  change  in  state  that 
takes  place  in  it,  so  any  electrode-potential  is  determined  solely  by  the 
change  in  state  that  takes  p.ace  at  the  electrode.  With  the  aid  of  this 
principle  and  the  general  expressions  for  free-energy-decrease  derived 
in  Arts.  127  and  129,  the  change  of  an  electrode-potential  with  the 
concentration  of  the  ions  or  with  that  of  any  other  gaseous  or  dis- 
solved substance  involved  in  the  electrode-reaction  can  readily  be 
calculated,  provided  the  concentrations  are  so  small  that  the  dissolved 
substances  conform  to  the  laws  of  perfect  solutes. 

Pro&.  29.  Derive  an  algebraic  expression  for  the  difference  EX  —  EX 
between  the  electrode-potentials  of  the  two  half-cells  involved  in  each 
of  the  following  cells,  by  considering  the  change  in  state  that  occurs  when 
one  faraday  is  passed  through  the  cell  and  assuming  that  the  dissolved 
substances  behave  as  perfect  solutes  up  to  a  concentration  of  1-molal  : 


ELECTRODE-POTENTIALS  163 

a.  Oa(patm.),  OH-(cm.),  OH-(lm.),  O2(latm.). 

the  two  electrode-potentials  involved  in  each  of  the  following  cells  : 

c.  Hg  +  HgaS04,  S04=(cm.),  SO4=(lm.),  Hg2S04  +  Hg. 

(j.  Ag-f  AyCl,  01-  (cm.),  Cl-(lm.),  AgCl  -f-  Ag. 

e.  CZa(patm.),  Cl-(cm.),  Cl-(lm.),  C72Clatm.). 

Pro&.  5(7.  Derive  an  algebraic  expression  for  the  difference  between 
a.  Ag,  Ag+(cm.),  Ag+(l  m.),  Ag.  b.  Zn,  Zn+  +  (c  in.),  Zir  +  U  m.  ),  Zn. 

c.  Hg,Hga++(cm.),Hg!l++(lm.),Hg.     d.     I,,  I-(om.),  I-(l  m.),  I* 

d.  M,  Cl,(clm.)+Cl-(c1m.),  Cl2(lm.)  +  Cl-(lm.),  M. 

e.  M,  Fe++(Cim.)-hFe+++(c2m.),  Fe++(l  m.)-|-  Fe+++(lm.),  M. 

The  electrode-potential  calculated  as  in  the  preceding  problems 
for  the  case  that  the  concentrations  of  the  ions  or  other  solutes  in- 
volved in  the  electrode-reaction  are  1  molal  and  the  pressure  is  one 
atmosphere  is  called  the  specific  electrode-potential.  It  will  be  here 
represented  by  the  symbol  E  followed  by  a  subscript  showing  the  ele- 
ment involved  in  the  electrode-reaction,  or,  when  necessary  for  clear- 
ness, by  a  parenthesis  showing  both  substances  involved  in  that 
reaction;  for  example,  by  such  symbols  as  EZn,  EH,  £(1%,  01"),  S(Fe+% 
tV++),  E(Ag  +  AgCl,  01").  Its  value  is  not  that  actually  observed 
when  the  concentrations  are  1  molal,  owing  to  the  large  deviation 
(discussed  in  Art.  86)  of  the  activity  of  ions  from  that  of  perfect 
solutes  at  such  high  concentrations.  It  is  to  be  regarded  as  an  em- 
pirical constant  derived  from  measurements  at  small  concentrations, 
from  which  conversely  the  true  electrode-potential  at  any  small  con- 
centration can  be  calculated. 

It  is  evident  from  the  preceding  problems  that  in  general  the 
electrode-potential  E(A,  B,  .  .)  when  the  concentrations  of  the  solutes 
A,  B,  .  .  involved  in  the  electrode-reaction  are  c  c  ,  .  .  is  related  to 
the  specific  electrode-potential  E(A,  B,  .  .)  in  the  way  expressed  by  the 
equation  : 


E(A,B;.  .)  =E(A,  B,  .  .)  -XAlogcA-N  log  CB  -  .. 

where  N^,  N^  .  .  represent  the  number  of  mols  of  the  solutes  A,  B,  .  . 
which  are  produced  when  one  faraday  of  positive  electricity  passes 
from  the  electrode  to  the  solution,  the  value  being  given  a  negative 
sign  for  any  solute  which  is  destroyed  instead  of  being  produced. 
When  one  of  the  substances  involved  in  the  electrode-reaction  is  a  gas 
its  concentration  may  be  replaced  by  its  pressure,  the  specific  electrode- 


164  ELECTROCHEMISTRY 

potential  in  that  case  being  the  electrode-potential  when  the  gas  is  at 
a  pressure  of  one  atmosphere.  For  convenience  in  numerical  calcula- 
tions it  may  be  noted  that  the  constant  2.303  R/F  (equal  to  2.303  X 
8.32/96500)  has  the  value  1.984  X  10"4. 

Prob.  31. — Determination  of  Specific  Electrode-Potentials. — Calculate 
the  specific  electrode-potentials  at  25°,  referred  to  the  calomel-electrode, 
of  a,  Hz,  H* ;  b,  C12  Cl~  ;  c,  Ag  -f  AgCl,  01-  ;  d,  Ag,  Ag-1-.  For  the  electro- 
motive-force values  needed  see  Probs.  28  and  21.  The  ionization  of  HC1 
at  0.1  normal  is  92%.  The  solubility  of  silver  chloride  in  water  at  25°  is 
1.30  X  !0~6  normal. 

The  specific  electrode-potential  of  Hv  H*  referred  to  the  calomel- 
electrode  has  been  calculated  in  the  preceding  problem.  From  data 
for  similar  cells  with  more  dilute  HC1  solutions  its  value  has  been 
more  accurately  determined,  and  found  to  be  +0.277  volt  at  25°.  This 
potential  is  evidently  the  electrode-potential  of  the  molal  hydrogen- 
e'ectrode,  Z7,(l  atm.),  H+(l  m.),  mentioned  in  Art.  133  as  one  of  the 
standard  half -cells.  It  is,  to  be  sure,  not  the  actual  potential  of  this 
half -cell;  but  it  is  the  potential  that  is  calculated  for  it  by  the  usual 
logarithmic  formula  from  the  electrode-potential  of  any  half-cell 
#2(latm.),  H+(cm.)  in  which  the  concentration  c  is  very  small. 

This  molal  hydrogen-electrode  will  be  used  as  the  standard  of 
reference  throughout  this  book,  because  of  its  simple  theoretical 
significance.  Its  potential  at  25°  is  substantially  identical,  so  far  as 
can  be  judged  from  existing  data,  with  that  of  the  half -cell  £T,(1  atm.), 
H,SO4(2n.),  which  is  often  employed  as  the  standard  hydrogen- 
electrode.  The  value  of  any  electrode-potential  at  25°  referred  to  this 
standard  is  evidently  0.277  volt  smaller  than  the  value  of  the  same 
electrode-potential  referred  to  the  calomel-electrode. 

135.  Values  of  the  Specific  Electrode-Potentials.— The  following 
table  contains  some  of  the  more  accurately  determined  values  of  the 
specific  electrode-potentials  at  25°   and  one  atmosphere,  referred  to 
that  of  the  molal  hydrogen-electrode  taken  as  zero. 
Li,  Li*         3.027        Cu+,  Cu*+  —0.20  Fe++,  Fe*++          —0.740 

K,K>  2.931         Ag,  AgCl -f  01-        —0.224          Ag,  Ar  —0.793 

Na,  Na+      2.721        Hg,  HgjClj-f  Cl~      — 0.270         Hg,  Hg/+  —0.80 

Zn,  Zn++     0.76  Cu,  Cu++  — 0.34  Hg/+,  Hg**          —0.92 

Fe,  Fe++      0.43  O2,  OH-  — 0.393         Br2  ( 1  in. ) ,  Br-     —1.10 

Cd,  Cd++     0.40          Cn,  Cu+  —0.51  C12,  01-  —1.357 

Pb.  Pb+*     0.13  I2, 1-  —0.533         C12, 01-  —1.388 


ELECTRODE-POTENTIALS  165 

Specific  electrode-potentials  which  are  related  to  one  another  can 
be  calculated  in  the  ways  illustrated  by  the  following  problems. 

Prol).  32.  State  what  data  would  be  needed  to  calculate  one  of  the 
following  specific  electrode-potentials  from  the  other,  and  formulate  an 
expression  by  which  the  calculation  could  be  made :  a,  C12,  Cl~  from  Clv 
Cl- ;  6,  Hg,  Hg2Cl2  +  Cl-  from  Hg,  Hg2++. 

Relation  between  the  Specific  Electrode-Potentials  of  an  Element 
that  exists  in  More  than  Two  States  of  Oxidation. — 

Prob.  38.  By  considering  the  effect  of  passing  electricity  through  the 
cells  Cu,  Cu+(lm.)  Cu+  +  (lrn.),  Cu,  and  Cu,  Cu++(l  m.)  1 1  Cu+(lm.) 
-fCu++(lm.),M,  derive  a  relation  between  the  specific  electrode-poten- 
tials of  Cu,  Cu+;  Cu+,  Cu++;  and  Cu,  Cu++;  and  calculate  the  last  of  these 
potentials  from  the  first  two. 

Pro 6.  33 A.  a.  By  considering  appropriate  cells  derive  an  expression 
for  the  specific  electrode-potential  of  Hg,  Hg*+  in  terms  of  those  of 
Hg,  Hga++  and  of  Kg/*,  Hg++.  b.  Calculate  the  value  of  the  first  of  these 
potentials  from  the  values  of  the  other  two  which  are  given  in  the  pre- 
ceding table. 

136.  Calculation  of  the  Electromotive  Force  of  Cells  from  the 
Specific   Electrode-Potentials. — The   electromotive  force   of   cells   in 
which  the  liquid-potentials  are  negligible  or  are  to  be  disregarded 
can  be  calculated  from  the  specific  electrode-potentials  in  the  way 
illustrated  by  the  following  problems. 

Prob.  34.  Calculate  the  electromotive  force  at  25°  of  the  cell 
Zn,  ZuCl2(  0.001  f.),  AgCl  +  Ag.  Consider  the  ionization  of  the  zinc 
chloride  to  be  complete. 

Pro  b.  35.  Calculate  the  electromotive  force  at  25°  of  the  cell 
H, (0.1  atm. ),  H2S04 (0.05  f.),  PbS04 -f  Pb.  The  saturated  solution  of 
PbSO4  at  25°  is  0.00014  formal.  In  regard  to  the  ionization  of  sulphuric 
acid  see  Prob.  38.  Art.  60. 

Prol).  36.  Formulate  numerical  expressions  for  calculating  the  elec- 
tromotive force  at  25°  of  each  of  the  following  cells : 

a.  M-f  CuCl,  CuCl2(0.01f.),  C72(0.1atm.). 

b.  M,  0.01  f.  KI  sat.  with  I2 1 1  FeCl2(0.02  f.)  4.  FeCl8(0.01  f.),  M. 
Regard  the  ionization  of  all  the  salts  as  complete.    The  solubility  of  CuCl 
is  0.0010  formal.    For  the  composition  of  iodide  solutions  saturated  with 
iodine  see  Prob.  52.  Art.  92. 

137.  Evaluation  of  Liquid-Potentials.— Just  as  the  electromotive 
force  of  a  whole  cell  is  determined  by  the  free-energy-decrease  attend- 
ing the  change  in  state  that  takes  place  in  the  cell,  and  just  as  the 
electrode-potential  is  determined  by  the  free-energy-decrease  attend- 


166  ELECTROCHEMISTRY 

ing  the  change  in  state  that  takes  place  at  the  electrode,  so  a  liquid- 
potential   is   determined   by   the   free-energy-decrease   attending   the 
change  in  state  that  takes  place  at  the  boundary  between  the  two 
-  solutions. 

The  change  in  state  at  such  a  boundary  is  of  a  simple  character 
in  the  case  where  the  two  solutions  contain  only  the  same  solute  at 
two  different  concentrations,  as  in  the  combination,  NaCl(0.01  f.), 
NaCl(0.1f.).  In  this  case  the  calculation  is  based  on  the  considera- 
tion of  the  number  of  equivalents  of  the  positive  ion-constituent  and 
of  the  negative  ion-constituent  that  pass  through  the  boundary  in 
the  two  opposite  directions  per  faraday  passed  through.  It  therefore 
involves  a  knowledge  of  the  transference-numbers  of  the  ions.  It  a^o 
involves,  since  the  free-energy-decrease  attending  the  transference  of 
the  ions  must  be  evaluated,  a  knowledge  of  the  ion-concentrations 
in  the  two  solutions. 

Prob.  87.  Calculate  the  liquid-potential  at  18°  of  the  combinations: 
a,  NaCl(0,lf.),  NaCl(O.Olt)  ;  6,  K2SO4(0.1  f.),  KjSCMO.Ol  f.).  For  the 
data  needed  see  Arts.  53  and  56. 

*By  a  similar  consideration  of  the  ion-transference  that  takes 
place  at  the  boundary  of  the  two  solutions  an  expression  can  be  derived 
for*  the  liquid-potential  of  combinations  of  the  types  KC1,  KOH; 
KC1,  NaCl;  K,SO4,  Na2S04;  ZnSO4,  CuS04,  for  the  case  that  the  two 
solutes  have  the  same  concentration  and  ionization.  Namely,  taking 
as  a  specific  example  the  combination  KCl(c  formal),  NaCl(c  formal), 
it  will  be  noted  that  there  must  be  at  the  boundary  a  portion  of  the 
solution  in  which  the  two  salts  are  present  in  varying  proportions, 
the  concentration  of  the  potassium  chloride  decreasing  continuously 
from  left  to  right  from  c  to  0,  and  that  of  the  sodium  chloride  increas- 
ing continuously  in  the  same  direction  from  0  to  c.  Representing  now 
by  T  and  T  the  transference-numbers  of  the  respective  ions  in  the 
two  solutions  of  the  pure  salts,  it  is  evident  that  per  faraday  of 
electricity  T  equivalents  of  potassium-ion  enter  the  left-hand  side  of 
the  boundary-portion,  and  that  no  potassium-ion  leaves  the  right-hand 
side  of  that  portion ;  that  no  sodium-ion  enters  the  left-hand  side,  but 
that  T  equivalents  of  sodium-ion  leave  the  right-hand  side ;  and  that 

t93> 

1 — T     equivalents  of  chloride-ion  leave  the  left-hand  side  and  1 — T  Na 

"This  paragraph  and  the  problem  following  it  may  be  omitted  in  brief 
courses. 


LIQUID-POTENTIALS  107 

equivalents  of  chloride-ion  enter  the  right-hand  side  of  the  boundary- 
portion.  The  resultant  change  in  state  attending  the  passage  of  one 
faraday  is  therefore  the  transfer  of  T  equivalents  of  KC1  from  the 
pure  potassium  chloride  solution  (where  its  concentration  is  c)  into 
the  boundary-portion  (where  its  concentration  has  values  varying  from 
c  to  0),  and  the  transfer  of  T  equivalents  of  NaCl  from  the  boundary- 
portion  (where  its  concentration  has  values  varying  from  0  to  c)  into 
the  pure  sodium  chloride  solution  (where  its  concentration  is  c).  The 
free-energy-decrease  attending  these  transfers  is  in  this  case  an 
integral  of  an  infinite  number  of  infinitesimal  free-energy-changes. 
By  evaluating  it  and  placing  it  equal  to  the  electrical  work  that 
attends  these  transfers  when  they  are  brought  about  by  the  passage 
of  the  current,  an  expression  for  the  liquid-potential  is  obtained. 

Pro ft.  38.  Describe  in  detail  the  change  in  state  that  takes  place 
when  one  faraday  passes  through  the  combination  Cu(NO,)a(0.01  f.), 
Zn(NO8)2(0.01f.). 

The  general  expression,  obtained  in  the  way  just  described,  for 
the  liquid-potential  E  between  solutions  of  any  two  substances  of  the 
same  ionic  type  having  one  ion  in  common  and  having  equal  concen- 
trations and  ionizations  is: 

EL  =  RT-      A,, 

"      ?  A02 

In  this  expression  A01  and  A02  denote  the  equivalent  conductances  at 
zero  concentration  of  the  left-hand  and  right-hand  solutes,  respectively, 
and  v  denotes  the  number  of  positive  charges  on  the  ions  not  common 
to  the  two  solutes,  it  being  given  a  negative  value  when  the  charges 
are  negative.  In  the  derivation  of  this  equation  it  is  assumed  that 
intermediate  or  complex  ions  are  not  present,  and  that  the  simple  ions 
behave  as  perfect  solutes  and  have  the  same  relative  conductances  up 
to  the  concentration  involved. 

Pro  ft.  S9.  Calculate  the  liquid-potential  at  18°  of:  a,  KOH(0.1f.), 
KC1(0.1  f.)  ;  6,  HC1(0.01  f.).  KC1(0.01  f.)  ;  c,  ZnSO«(0.1  f.),  CuSO4(0.1  f.). 

Pro  ft.  40.  Calculate  the  electromotive  force  at  18°  of  the  cell 
Ag,  AgN03(0.1n.),  KNOs(0.1n.),  KCl(0.1n.),  KC1(1  n.)  -f  Hg,Cl2,  Hg. 
Assume  that  the  potassium  chloride  is  74%  ionized  at  1  normal  and 
84%  ionized  at  0.1  normal,  and  that  the  other  two  salts  are  84%  ionized 
at  0.1  normal. 


168  ELECTROCHEMISTRY 

For  combinations  of  solutions  of  two  salts  having  a  common  ion 
but  different  concentrations,  such  as  KNO3(0.1  n.),  KCXO-Oln.),  and 
for  combinations  of  solutions  of  salts  without  a  common-ion,  such  as 
KN"O3(0.1n.),  NaCl(0.01n.),  the  calculation  of  the  liquid-potential 
is  very  complicated.  Combinations  of  these  kinds  can,  however,  be 
avoided,  as  illustrated  by  the  cell  of  Prob.  40,  by  connecting  the  two 
solutions  through  intermediate  ones  so  as  to  produce  only  combina- 
tions of  the  two  types  for  which  the  liquid-potentials  can  be  calculated 
as  described  above. 

It  is  to  be  noted  that  the  above-derived  expressions  for  the  liquid- 
potentials  involve  the  assumption  that  the  ions  behave  as  perfect 
solutes,  and  therefore  that  these  expressions  are  exact  only  at  small 
concentrations.  The  assumption  is  also  involved  in  the  case  of  tri- 
ionic  salts  that  these  are  ionized  only  into  the  ultimate  ions  (such  as 
K+  and  SO4=),  without  formation  of  an  appreciable  proportion  of  the 
intermediate  ion  (such  as  KSO4~). 

A  liquid-potential  can  sometimes  be  experimentally  determined  by 
measuring  the  electromotive  force  of  a  cell  involving  it  in  which  the 
two  electrode-potentials  are  made  substantially  equal,  as  in  the  cells: 
Hg  +  Hg2Cl2,  KCKO.lf.),  NaCKO.lf.),  Hg2CL  +  Hg. 
Ag,  AgNO3 (0.001  f.)  +  KNO3(0.1  f.),  AgNO3  (0.001  f.)  + 

NaNO:!(0.1f.),  Ag. 

ProJ).  41.  a.  Show  that  the  electromotive  force  of  the  silver  cell  here 
formulated  is  substantially  equal  to  the  liquid-potential  of  KNO3(0.1f.), 
NaNO8(0.1f.),  by  specifying  exactly  what  determines  each  of  the  three 
partial  potentials  of  the  cell.  ft.  Show  why  there  would  be  a  considerable 
error  in  deriving  the  liquid-potential  of  KNO3 ( 0.1  f.),  NaNO3( 0.01  f.) 
from  the  electromotive  force  of  a  cell  which  differs  from  the  one  just 
considered  only  in  the  respect  that  the  NaNO3(0.1f.)  is  replaced  by 
NaN03(0.01f.). 

138.  Determination  of  Ion-Concentrations  and  of  Equilibrium- 
Constants  Involving  Them  by  Means  of  Electromotive-Force  Meas- 
urements. 

Prol).  4%- — Determination  of  Solubility. — The  electromotive  force  of 
the  cell  Ag  +  Agl,  KI(0.1f.),  KNO8(0.1f.),  AgNO3  (0.1  f. ) ,  Ag  is  0.814 
volt  at  25°.  Calculate  the  solubility  of  silver  iodide  in  water  at  25°. 
Assume  the  ionizations  of  all  the  substances  to  be  equal  and  to  have  the 
average  value  shown  by  salts  of  the  uniunivalent  type  (given  in  Art.  58). 


EQUILIBRIUM  OF  OXIDATION  REACTIONS  169 

Prob.  43. — Determination  of  the  lonization-Constant  of  Water.— 
Calculate  the  ionization-constant  of  water  (Art.  85)  at  18°  from  the 
fact  that  the  electromotive  force  of  the  cell  fT2(latm.),  HCl(0.1f.), 
KCl(0.1f.),  KOH(0.1f.),  #2(latm.)  is  —0.653  volt  at  18°.  Make  the 
same  assumption  in  regard  to  the  ionizations  as  in  the  preceding  problem. 

Prob.  44. — Determination  of  Complex-Constants. — The  electromotive 
force  of  the  cell  Ag,  K+Ag(CN)2-(0.01  n.)  -fKCN(ln.),  KCl(ln.)t 
Hg2Cl2-fHg  is  0.83  volt  at  25°.  Calculate  the  dissociation-constant  of 
the  complex-ion  Ag(CN)2~  at  25°.  Assume  the  ionizatkm  of  the  salts  to 
be  74%,  and  neglect  the  liquid-potential,  which  is  small  in  this  case. 

Prob.  45. — Determination  of  the  Hydrolysis  of  Salts. — When  a  solu- 
tion 0.05  formal  in  Na2HPO4  is  saturated  with  hydrogen  at  1  atm.,  when 
a  platinum  electrode  is  placed  in  it,  and  when  the  half-cell  HCl(0.01f.) 
-f-  NaCl  (0.1  f.),  H2(~L  atm.),  is  brought  into  contact  with  it,  the  cell  thus 
formed  is  found  to  have  at  18°  an  electromotive  force  of  0.398  volt. 
Calculate  the  hydrolysis  of  the  salt,  assuming  complete  ionization  of  the 
largely  ionized  substances,  and  neglecting  the  liquid-potential,  which  is 
made  small  by  the  addition  of  the  sodium  chloride. 

Prob.  46. — Determination  of  Indicator-Constants. — When  the  Na,HPO4 
solution  of  Prob.  45  is  made  0.0001  formal  in  phenolphthalein  the  indi- 
cator is  found  to  show  a  color  13%  as  intense  as  that  which  is  pro- 
duced on  adding  to  the  solution  an  excess  of  sodium  hydroxide.  Calculate 
the  ionization-constant  of  this  indicator. 

139.  Derivation  of  the  Equilibrium- Conditions  of  Oxidation  Reac- 
tions from  the  Electrode-Potentials. — When  the  concentrations  of  the 
substances  involved  in  the  two  electrode-reactions  are  such  that 
the  two  electrode-potentials  are  equal  to  each  other,  there  is  evidently 
no  tendency  for  the  cell  to  act  nor  for  the  chemical  change  to  take 
place  in  it.  In  other  words,  the  concentrations  that  make  the  two 
electrode-potentials  equal  are  concentrations  at  which  the  chemical 
change  is  in  equilibrium.  The  equilibrium-constant  of  the  chem- 
ical change  which  is  the  resultant  of  the  two  electrode-reactions  can 
therefore  be  calculated  from  the  specific  electrode-potentials. 

Prob.  4~-  Calculate  the  concentration  of  copper-ion  at  which  the 
reaction  Zn  -(-  Cu++  =  Cu-f  Zn++  is  in  equilibrium  when  the  zinc-ion  is 
1  molal. 

Prob.  48.  a.  Calculate  the  concentration  of  hydrogen-ion  at  which 
the  reaction  Pb  -f  2H+C1-  =  H2  -f-  Pb++Cl-2  is  in  equilibrium  at  25°  when 
the  hydrogen  has  a  pressure  of  1  atm.  and  the  lead-ion  is  0.03  molal. 

b.  Formulate  an  algebraic  relation  between  the  equilibrium-constant  of 
the  corresponding  ionic  reaction  and  the  specific  electrode- potentials. 

c.  Calculate  the  value  of  this  equilibrium-constant. 


170  ELECTROCHEMISTRY 

Prob.  49 '•  «•  Formulate  an  algebraic  relation  between  the  equilibrium- 
constant  of  the  ionic  reaction  Ag-|- Fe*+*  — Ag+ -f- Fe++  and  the  specific 
electrode-potentials  involved.  b.  Calculate  the  value  of  this  equilibrium- 
constant,  c.  Calculate  the  composition  of  the  equilibrium-mixture  that 
results  when  metallic  silver  is  placed  in  0.1  formal  Fe(NO3)8  solution, 
assuming  complete  ionization  of  the  salts. 

Prob.  50.  Formulate  a  complete  numerical  expression  for  calculating 
the  OH~  concentration  at  which  1000  times  as  much  manganate-iou 
(MnO4=)  as  permanganate-ion  (MnO4-)  is  present  in  contact  with  air 
when  equilibrium  is  reached  at  25°.  The  specific  electrode-potential  of 
MuO4=,  MnO4-  has  been  roughly  determined  to  be  — 0.61  volt  at  25°. 

Prob.  51.  a.  Derive  an  expression  for  the  equilibrium-constant  of 
the  reaction  Cu  +  Cu++  =  2  Cu*  in  terms  of  the  specific  electrode-poten- 
tials. I).  Calculate  the  concentration  of  cuprous  salt  resulting  when 
copper  is  shaken  with  a  0.1  formal  CuSO4  solution  at  25°,  assuming 
complete  ionization.  'c.  Calculate  the  concentration  of  cupric  salt  in  the 
equilibrium-mixture  produced  by  shaking  a  0.1  formal  CuCl2  solution 
with  copper  at  25°. 

140.  Electromotive  Force  of  Cells  with  Concentrated  Solutions. — 

The  electromotive  force  of  any  cell  in  which  solutes  are  present  at 
large  concentrations  cannot  be  calculated  from  the  specific  electrode- 
potentials  with  the  aid  of  the  logarithmic  concentration  formula  of 
Art.  134,  since  this  holds  true  even  approximately  only  when  the 
concentration  does  not  exceed  about  0.1  normal.  A  knowledge  of  the 
change  in  state  taking  place  in  such  a  cell  and  of  the  reactions 
occurring  at  its  electrodes  is,  however,  of  importance,  since  it  shows 
qualitatively  the  factors  which  determine  the  magnitude  of  the  elec* 
tromotive  force.  This  is  illustrated  by  the  following  problems,  which 
relate  to  certain  cells  of  technical  importance. 

Prob.  52.  One  form  of  the  Clark  standard  cell  at  20°  is  repre- 
sented by  the  formula :  Zn  -f  ZnS047H20,  ZnSO416.8H8O,  Hg2S04  -j-  Hg. 
a.  Specify  the  exact  change  of  state  that  occurs  when  two  faradays 
pass  through  the  cell.  b.  Show  that  even  in  an  actual  cell,  with  only 
a  finite  quantity  of  solution,  no  variation  of  the  electromotive  force 
results  when  two  faradays  pass,  provided  the  change  in  state  takes 
place  slowly  enough. 

Prob.  58.  The  LeclanchS  cell  consists  essentially  of  a  zinc  rod 
(amalgamated  to  diminish  local  voltaic  action)  dipping  into  a  con- 
centrated NH4C1  solution  containing  ZnCl2,  and  a  carbon  rod  coated 
with  MnO2  dipping  into  the  same  solution,  a.  Formulate  the  ion-reac- 
tion that  takes  place  at  each  electrode  and  the  whole  reaction  in  the 
cell,  taking  into  account  the  facts  that  the  MnO2  is  reduced  to  Mn2O,, 


ELECTROMOTIVE  FORCE  OF  CELLS  171 

and  that  Zn(OH)2  is  soluble  in  NH4C1  solution.  6.  Show  in  what 
direction  the  electromotive  force  of  the  cell  would  be  changed  by 
decreasing  the  concentration  of  the  NH4C1. 

Note. — The  common  "dry  cell"  is  a  Leclanche"  cell  to  which  some 
porous  material,  such  as  paper-pulp  or  sawdust,  has  been  added  to  hold 
the  liquid. 

Prol.  54.  In  the  nickel-iron  (Edison)  storage  cell,  Fe,  KOH(21%), 
Ni(OH)3,  the  main  reaction  is  Pe  -{-  2Ni(OH),  —  Fe(OH),  -f  2Ni(OH)a 
(the  degree  of  hydration  of  the  three  oxides  being,  however,  somewhat 
indefinite),  a.  Write  the  ion-reaction  occurring  at  each  electrode. 
6.  Show  in  what  direction  each  of  the  electrode-potentials,  and  also 
the  electromotive  force  of  the  whole  cell,  would  vary  with  increase  of 
the  KOH  concentration. 


172  ELECTROCHEMISTRY 

VOLTAIC   ACTION.   ELECTROLYSIS,   AND    POLARIZATION 

141.  Concentration-Changes  attending  Voltaic  Action  and  the 
Resulting  Polarization. —  Throughout  the  foregoing  considerations, 
as  an  aid  in  evaluating  the  electromotive  force,  it  has  been  assumed 
that  so  large  a  quantity  of  solution  is  present  in  the  cell  that  only 
infinitesimal  concentration-changes  are  produced  in  it  by  the  passage 
of  a  finite  quantity  of  electricity.  The  fact  that  this  is  not  the  case 
in  actual  cells  must  be  taken  into  account. 

Prob.  55.  The  electromotive  force  E  at  15°  of  the  lead  storage-cell 
varies  with  the  mol-fraction  x  of  the  H2SO4  (for  values  of  x  up  to  0.10) 
according  to  the  equation  E  —  1.855  -f-  3.80#  —  10#2.  Out  of  a  cer- 
tain cell  which  contains  1300  g.  of  10  mol-percent  H2SO4  a  steady 
current  of  5.36  amperes  is  taken  for  10  hours,  a.  Calculate  the  electro- 
motive force  of  the  cell  at  the  beginning  and  at  the  end.  6.  Derive  an 
integral  (expressed  in  terms  of  molal  composition  and  numerical  co- 
efficients) by  which  the  electrical  energy  producible  by  the  outflow  of 
any  given  quantity  of  electricity  from  any  similar  lead  storage-cell 
could  be  calculated. 

In  practice  the  electromotive  force  is  often  decreased  much  more 
than  these  considerations  indicate  because  tho  concentration-changes 
actually  occur  in  the  immediate  neighborhood  of  the  electrodes  and 
are  only  gradually  distributed  by  convection  or  diffusion  through  the 
whole  body  of  the  solution.  Thus  in  the  lead  storage-cell  water  is 
produced  and  acid  is  destroyed  by  the  electrode-reaction  in  the  solu- 
tion impregnating  the  porous  lead-peroxide  electrode,  and  the  acid 
can  only  be  replenished  from  the  main  body  of  the  solution  by  the 
slow  process  of  diffusion.  This  phenomenon  i?  one  kind  of  polariza- 
tion, sometimes  called  concentration-polarization;  the  name  polar- 
ization being  used  in  general  to  denote  the  production  by  the  passage 
of  the  current  of  any  change  in  the  solution  adjoining  the  electrode 
or  in  the  surface  of  the  electrode  which  makes  its  potential  deviate 
from  its  normal  value. 

Prob.  56.  When  a  certain  current  is  taken  at  25°  out  of  a  certain 
cell  of  the  form  Zn,  ZnSO4(lf.),  CuSO4(lf.),  Cu,  the  electromotive 
force  soon  becomes  fairly  constant  and  remains  so  for  a  time  at  a  value 
0.06  volt  below  its  normal  value,  a.  Show  quantitatively  how  this 
might  be  accounted  for  by  concentration-polarization.  6.  Explain  how 
the  polarization  would  be  affected  by  increasing  the  current ;  by  using 


ELECTROLYSIS  AND  POLARIZATION  173 

smaller  electrodes  without  changing  the  current,  thereby  increasing  the 
current-density,  by  which  is  meant  the  current  in  amperes  per  unit- 
area  (per  square  centimeter  or  decimeter)  of  electrode-surface ;  and  by 
stirring  the  solutions,  these  being  separated  from  each  other  by  a 
porous  cup. 

142.  Electrolysis  in  Relation  to  Applied  Electromotive  Force. — 
In  order  to  produce  electrolysis  in  any  electrolytic  cell  there  must 
obviously  be  applied  from  an  external  source  an  electromotive  force 
a.t  least  equal  to  the  electromotive  force  that  is  produced  by  the 
combination  of  solution  and  electrodes,  considering  it  as  a  voltaic 
cell.  The  value  of  the  electromotive  force  that  must  be  applied  to 
compensate  this  ~back  electromotive  force  of  the  cell  under  the 
actual  conditions,  and  thus  produce  appreciable  electrolysis,  is  called 
the  decomposition-potential.  Its  value,  when  not  influenced  by  in- 
definite polarization-effects,  such  as  are  described  in  Art.  143,  can 
therefore  be  calculated  by  the  methods  considered  in  Arts.  130-137. 
It  is  important  to  recognize  this  fact;  for  the  decomposition-poten- 
tial is  sometimes  treated  as  if  it  were  an  essentially  independent 
quantity. 

Pro&.  57. — Experimental  Determination  of  Decomposition-Potential. 
— A  0.05  formal  solution  of  CuSO4  is  placed  at  25°  in  a  certain  electro- 
lytic cell  between  a  mercury  electrode  covered  with  Hg2SO4  and  a  cop- 
per electrode.  An  external  electromotive  force  is  applied  at  the  mercury 
electrode,  first  of  0.1,  then  of  0.2,  0.3,  0.4,  0.5,  and  0.6  volt.  An  ammeter 
placed  in  series  with  the  cell  shows  that  no  appreciable  current  passes 
until  0.4  volt  is  applied,  when  the  current  is  0.8  milliamperes ;  while 
with  0.5  volt  the  current  is  2.8  milliamperes,  and  with  0.6  volt  it  is 
4.8  milliamperes.  a.  Plot  these  values  of  the  current  as  ordinates  and 
of  the  potential  as  abscissas,  and  derive  from  the  plot  the  value  of  the 
decomposition-potential.  6.  Calculate  the  resistance  of  the  solution. 

ProT).  58.— Deposition  of  Metals  ~by  Electrolysis. — A  solution  0.05 
formal  in  H2SO4  and  0.05  formal  in  CuSO4  is  electrolyzed  at  25°  between 
a  mercury  anode  and  a  platinum  cathode.  In  the  mixture  the  salt  may 
be  assumed  to  be  35%  ionized  and  the  acid  to  be  35%  ionized  into  2H+ 
and  SO4=  and  60%  ionized  into  H+  and  HSO4-.  a.  Calculate  from  the 
electrode-potentials  (taking  that  of  Hg  4.  Hg2S04,  SO4=(lm.)  as  —0.62 
volt)  the  minimum  electromotive  force  which  would  have  to  be  applied 
in  order  to  cause  the  copper  to  deposit.  6.  To  what  value  would  this 
electromotive  force  have  to  be  increased  after  99%  of  the  copper  had 
been  precipitated  in  order  that  the  deposition  might  continue,  assuming 


174  ELECTROCHEMISTRY 

the  ionizations  to  be  the  same  as  in  the  original  mixture?  c.  What  is 
the  minimum  electromotive  force  at  which  hydrogen  could  be  continu- 
ously set  free,  assuming  that  it  attains  at  the  cathode  an  effective 
pressure  of  1  atm.? 

Prod.  59. —  Separation  of  Elements  by  Electrolysis. — A  solution 
0.1  normal  in  KC1,  0.1  normal  in  KBr,  and  0.1  normal  in  KI  is  placed, 
together  with  a  platinum  electrode,  in  a  porous  cup ;  and  this  is  placed 
within  a  larger  vessel  containing  a  zinc  electrode  and  a  large  quan- 
tity of  0.1  normal  ZnCl2  solution.  Assuming  complete  ionization  of  the 
salts,  calculate  the  applied  electromotive  force  required  at  25°,  a,  to 
liberate  99.9%  of  the  iodine ;  6.  to  set  bromine  free  at  a  concentration 
of  0.0001  molal;  c,  to  liberate  99.9%  of  the  bromine  (which  remains  in 
solution)  ;  and  d,  to  liberate  chlorine  at  a  concentration  of  0.0001 
molal. 

143.  Electrolysis  in  Relation  to  Polarization. — In  electrolysis,  as 
in  voltaic  action,  concentration-changes  are  produced  at  the  elec- 
trodes; and  these  have  the  effect  of  increasing  the  applied  electro- 
motive force  required.  Thus  in  electrolyzing  a  solution  of  copper 
sulphate  between  copper  electrodes  (as  is  done  in  copper  plating) 
only  an  infinitesimal  electromotive  force  is  required  to  start  the 
deposition;  but  the  solution  around  the  cathode  soon  becomes  less 
concentrated  in  copper  and  the  solution  around  the  anode  more  con- 
centrated in  copper,  producing  a  concentration-cell,  like  those  of 
Art.  131,  with  an  electromotive  force  opposite  to  that  applied.  This 
back  electromotive  force  can  of  course  be  calculated  for  any  given 
concentration-changes.  It  would  evidently  be  diminished  by  decreas- 
ing the  current-density,  by  agitating  the  whole  solution,  or  by 
rotating  the  electrodes. 

When  a  gas,  such  as  hydrogen  or  oxygen,  is  set  free  at  an  elec- 
trode, a  phenomenon,  known  as  gas-polarizaiion,  is  observed  which 
does  not  occur  in  the  deposition  of  metals.  For  when  a  gas  is  in- 
volved in  the  electrode-reaction  its  partial  pressure  determines  the 
electrode-potential.  Hence  in  a  cell  exposed  to  the  air  a  slow,  con- 
tinuous electrolysis  may  take  place  before  the  applied  electromotive 
force  equals  the  electromotive  force  of  the  cell  when  the  gas-pressure 
is  one  atmosphere;  for  the  gas-forming  substance  is  produced  on 
the  electrode  at  a  lower  pressure  and  is  dissolved  in  the  solution  and 
carried  away  by  convection  and  diffusion.  Moreover,  a  sudden  in- 


ELECTROLYSIS  AND  POLARIZATION  175 

crease  in  the  rate  of  electrolysis  does  not  take  place  when  the  applied 
electromotive  force  is  increased  beyond  that  corresponding  to  a  pres- 
sure of  one  atmosphere;  for  the  gas-forming  substance  then  passes 
into  the  electrode-surface  at  this  high  pressure,  producing  a  super- 
saturated solid  solution  from  which  the  gas  does  not  escape  rapidly 
enough  to  reduce  the  effective  pressure  to  one  atmosphere.*  There 
is  therefore  a  back  electromotive  force  produced  by  the  cell  which 
is  larger  than  its  electromotive  force  when  the  gas-pressure  is  one 
atmosphere.  The  excess  of  the  one  over  the  other  is  called  the 
polarization  or  overvoltage. 

Proft.  60. —  Concentration-Polarization. — Calculate  the  electromotive 
force  of  polarization  that  would  result  in  electrolyzing  at  25°  a  0.01 
formal  CuSO4  solution  between  copper  electrodes  if  the  copper-ion  con- 
centration became  0.001  formal  around  the  cathode  and  0.1  formal 
around  the  anode,  assuming  equal  ionization  of  the  salt  at  these  two 
concentrations. 

Prob.  61. —  Gas  Polarization. — Electromotive  forces  successively 
increasing  in  magnitude  were  applied  at  22°  to  an  electrolytic  cell  con- 
sisting of  a  large,  unpolarizable,  platinized  platinum  plate  as  anode, 
a  small  mercury  surface  as  cathode,  and  a  0.1  normal  H2SO4  solution 
as  electrolyte.  Hydrogen  gas  at  1  atm.  was  bubbled  steadily  through 
the  cell,  and  a  resistance  of  100,000  ohms  was  placed  in  series  with  it, 
the  resistance  of  the  cell  being  negligible  in  comparison.  The  current- 
strengths  i  in  millionths  of  an  ampere  corresponding  to  various  applied 
electromotive  forces  E  in  volts  were  as  follows: 

i 0.06        0.44        1.20        2.20        3.70        4.82 

E 0.32        0.48        0.62        0.77        0.95         1.08 

a.  Plot  these  current-strengths  as  ordinates  against  the  electromotive 
forces  as  abscissas.  6.  Calculate  the  back  electromotive  force  corre- 

*  In  some  cases  the  solid  solution  is  probably  a  solution  in  the  metal  of 
the  gas-forming  substance  itself  or  of  the  corresponding  monatomic  substance 
(thus  of  H  or  O  in  the  case  of  H2  or  O2).  In  other  cases  there  is  evidence  that 
it  is  a  solution  in  the  metal  of  an  unstable  compound  of  the  metal  with  the 
gas-forming  substance  ;  thus  higher  oxides  such  as  PtO8  and  NiO3  seem  to  be 
formed  when  oxygen  is  set  free  on  platinum  or  nickel.  Such  compounds  have 
a  high  dissociation-pressure  and  are  therefore  equivalent  in  their  effect  on  the 
potential  to  the  gas  Itself  at  the  same  high  pressure,  assuming  equilibrium  to  be 
attained  between  them  and  the  electrolyte  in  the  solution. 

The  slowness  with  which  the  gas  escapes  from  these  solid  solutions  may 
arise  in  part  from  the  small  rate  at  which  the  reactions  (such  as  2H  =  H2, 
or  2PtO3  =  2Pt  +  3O2)  take  place  within  the  electrode-surface.  The  rate  of 
escape  of  the  gas  is,  moreover,  doubtless  largely  determined  by  surface-tension 
effects. 


176  ELECTROCHEMISTRY 

spending  to  each  of  these  applied  electromotive  forces,  record  these 
values,  and  plot  the  current-strengths  against  them  on  the  same  dia- 
gram, c.  Calculate  the  effective  pressure  of  the  hydrogen  at  the  elec- 
trode corresponding  to  the  smallest  of  these  back  electromotive  forces. 

The  back  electromotive  force  is  experimentally  determined  by  the 
method  illustrated  in  Probs.  57  and  61,  or  by  finding  the  smallest 
value  of  the  applied  electromotive  force  at  which  bubbles  form 
slowly  but  continuously  at  the  electrode  surface  (this  giving  a  value 
corresponding  practically  to  a  minimum  current-density),  or  by 
applying  a  definite  electromotive  force  to  the  cell  long  enough  to 
charge  the  electrodes  with  the  decomposition-products  and  then  short- 
circuiting  the  electrodes  through  a  high-resistance  potentiometer,  the 
applied  potential  being  at  the  same  time  removed.  From  this  back 
electromotive  force  the  overvoltage  is  obtained  by  subtracting  the 
theoretical  electromotive  force  corresponding  to  a  gas-pressure  of  one 
atmosphere. 

The  overvoltage  is  always  found  to  increase  with  increase  of  the 
applied  electromotive  force  and  of  the  current-density;  but  it  varies 
in  a  highly  specific  way  with  the  chemical  nature  of  the  gas  and 
with  the  chemical  nature  and  physical  state  of  the  metallic  electrode. 
Thus  in  certain  experiments  made  with  1-normal  H2SO4  with  a  low 
applied  potential  and  low  current-density  the  overvoltage  of  the 
hydrogen  was  found  to  be  zero  on  a  platinized  electrode,  0.03  volt 
on  smooth  platinum,  0.36  volt  on  lead,  and  0.44  volt  on  mercury; 
while  with  a  much  higher  applied  potential  and  current-density  it 
was  found  to  be  0.07  volt  on  platinized  platinum,  0.65  volt  on  smooth 
platinum,  1.23  volts  on  rough  lead,  and  1.30  volts  on  mercury.  And 
in  certain  experiments  with  2-normal  KOH  with  a  moderate  cur- 
rent-density the  overvoltage  of  the  oxygen  was  found  at  the  start 
to  be  0.44  volt  on  platinized  platinum,  0.84  volt  on  smooth  platinum, 
and  0.50  von  on  iron;  the  value  increasing  to  1.46  on  smooth  plat- 
inum and  to  0.59  on  iron  after  two  hours'  passage  of  the  current. 

The  phenomenon  of  gas-polarization  and  the  overvoltage  attend- 
ing it  are  of  great  significance  in  technical  processes.  The  over- 
voltage  may  greatly  diminish  the  energy-efficiency  of  the  process, 
the  energy-efficiency  being  the  ratio  of  the  minimum  electrical 
energy  theoretically  required  to  produce  a  definite  quantity  of  some 


ELECTROLYSIS  AND  POLARIZATION  177 

product  of  the  electrolysis  to  the  energy  actually  expended.  Over- 
voltage  may  also  make  processes  practicable  which  would  otherwise 
not  be  possible;  thus  in  charging  a  lead  storage-cell  hydrogen  is  not 
set  free  at  the  lead  electrode  owing  to  its  overvoltage,  although  the 
potential  of  the  half-cell  jff2(latm.),  H2SO420H2O  is  about  0.4  volt 
less  than  that  of  the  half-cell  Pb  -f  PbS04,  H2SO420H20. 

Pro&.  62. — Energy-Efficiency  and  Overvoltagei — In  a  certain  com- 
mercial alkali-chlorine  cell  sodium  hydroxide  and  chlorine  are  produced 
at  an  iron  cathode  and  graphite  anode,  respectively,  by  the  continuous 
electrolysis  of  a  25%  NaCl  solution  which  is  slowly  flowed  through  a 
diaphragm  from  the  anode  to  the  cathode  compartment  (to  prevent 
hydroxide-ions  from  migrating  to  the  anode).  In  practice  4.5  volts  are 
applied  to  the  cell,  whose  resistance  is  0.00080  ohm,  yielding  a  current 
of  2000  amperes.  There  flow  off  each  hour  from  the  cathode  27,460  g.  of 
solution  containing  10%  NaOH  and  13%  NaCl.  The  electromotive  force 
of  the  voltaic  cell  #2(latm.  ),  NaOH  ( 10% ) -f  NaCl  (13%),  NaCl  (25%), 
C7,(latm.)  has  been  independently  determined  to  be  2.3  volts,  a.  Cal- 
culate the  current-efficiency  in  the  production  of  the  sodium  hydroxide. 
&.  Calculate  the  energy-efficiency,  c.  Calculate  the  overvoltage  and  the 
percentage  loss  in  energy-efficiency  that  arises  from  it. 

144.  Review  Problems. 

Pro&.  63.  The  potential  of  a  platinum  electrode  in  an  acid  solution 
of  potassium  iodate  and  iodine  is  found  to  depend  only  on  the  concen- 
trations of  H+,  IO3-,  and  I2.  a.  Write  the  electrode-reaction  attending 
the  passage  of  ten  faradays  from  the  electrode  to  the  solution.  6.  For- 
mulate an  expression  by  which  the  electrode-potential  E  at  25°  for  any 
small  concentrations  (H+),  (I<V),  (I.)  could  be  calculated  from  the 
electrode-potential  E'  for  the  case  that  (H+)  —  0.1,  (IO8-)  =  0.05,  and 
(I,)  =  0.02  formal. 


CHAPTER  X 

THERMODYNAMIC  CHEMISTRY:    THE  PRODUCTION   OF 

WORK  BY  CHEMICAL  CHANGES  AND  ITS  RELATION 

TO  THEIR  EQUILIBRIUM  CONDITIONS 


FREE-ENERGY     CHANGES    ATTENDING     ISOTHERMAL     CHANGES 

IN    STATE 

145.  Chemical  Significance  of  Free  Energy. — It  was  shown  in  the 
preceding  chapter  that  a  knowledge  of  the  maximum  work  produced 
by  isothermal  changes  in  state,  or  of  the  corresponding  free-energy- 
decrease  attending  them,  enables  the  electromotive  force  of  voltaic 
cells  to   be  evaluated.    It  will  be  shown  in  this  chapter  that   this 
knowledge  also  makes  possible  the  determination  of  the  equilibrium- 
conditions  of  chemical  changes.    The  development  of  the  principles 
relating  to  free  energy  and  the  working-out  of  a  complete  system 
of    free-energy    values    are    therefore    problems    of    great    chemical 
importance. 

The  principles  determining  the  free-energy-decrease  attending 
certain  kinds  of  changes  have  been  considered  in  the  preceding 
chapter.  These  are  summarized  in  Art.  147.  Other  principles  will 
be  described  in  later  articles.  First,  however,  there  will  be  presented 
in  the  next  article  the  conventions  adopted  in  this  book  for  the 
expression  of  free-energy  values. 

146.  Conventional   Expression   of  Free-Energy   Values.— Little 
more  need  be  said  in  regard  to  the  expression  of  free-energy  values 
than  that  the  same  conventions  that  have  already  been  described  in 
Art.    113   for  the   expression   of   heat-effects   will   be   employed   for 
expressing    free    energies.     When    necessary,    equations    expressing 
changes  in  free  energy  may  be  distinguished  from  those  expressing 
changes  in  heat-content  by  prefixing  to  them  the  symbols   (F)  and 
(H),  respectively;    and  the  absolute  temperature  may  be  shown  by 
attaching  to  this  letter  a  subscript.    For  example: 

(F^)  H2(l  atm.)  -f  i#2(l  atm.)  =  HaO  +  56,620  cal. 
(H«)  JET.d  atm.)  +  i0f(l  atm.)  =  H2O  +  68,400  cal. 
(F^)  H2O  =  —56,620  cal. 

The  pressure  must   always  be   definite;    when  not  specified,   it   is 

179 


180  THERMODYNAMIC  CHEMISTRY 

understood  to  be  one  atmosphere.  Throughout  this  chapter  free- 
energy  values  will  be  expressed  in  calories  on  account  of  their  close 
relations  with  heat-content  data,  which  are  commonly  so  expressed. 

In  accordance  with  the  specified  convention  as  to  the  arbitrary 
zero-point  of  the  scale,  the  free  energy  at  any  definite  temperature 
of  any  definite  quantity  of  a  substance  in  any  definite  state  is  equal 
to  the  free-energy-increase  that  attends  the  formation  of  the  sub- 
stance in  that  state  out  of  the  pure  elementary  substances  at  the 
same  temperature,  and  at  a  pressure  of  one  atmosphere,  each  ele- 
mentary substance  being  in  the  state  of  aggregation  that  is  the  stable 
one  at  this  temperature  and  pressure.  Thus  the  free  energy  of 
!#/(!  atm.)  refers  at  25°  to  its  formation  from  gaseous  hydrogen 
and  solid  iodine  at  one  atmosphere;  but  at  400°  it  refers  to  its 
formation  from  gaseous  hydrogen  and  gaseous  iodine  at  one 
atmosphere. 

The  free  energy  of  a  substance  in  a  solution  is  understood  to 
mean  the  free-energy-increase  attending  the  formation  of  the  pure 
substance  out  of  the  elementary  substances  plus  that  attending  the 
introduction  of  it  into  an  infinite  quantity  of  the  solution  under 
consideration.  This  free  energy  is  indicated  by  attaching  to  the 
chemical  formula  in  Roman  type  a  parenthesis  showing  the  concen- 
tration or  composition  of  the  solution;  for  example,  lKCl(at  O.lf.), 
lH2SO4(in  H2S0410H2O).  The  free  energy  of  water  in  a  solution 
is  often  conveniently  referred  to  that  of  pure  water  (instead  of  to 
that  of  hydrogen  and  oxygen)  as  zero;  and  when  so  referred 
the  symbol  .ZVAq  (instead  of  7VH2O)  is  used;  for  example, 
5Aq(in  H2SO410H2O). 

Prob.  1.  a.  Formulate  a  free-energy  equation  for  the  change  in 
state  considered  in  Prob.  17,  Art.  129.  6.  Find  the  free  energy  of 
lHCl(at  O.lf.)  at  25°.  c.  State  what  other  free-energy  change  would 
have  to  be  combined  with  this  one  in  order  to  give  the  free  energy  at 
25°  of 


147.  Tree-Energy  Principles  Already  Considered.—  It  was  shown 
in  Art.  125  that  the  maximum  work  (WR)  producible  by  any  iso- 
thermal change  in  state  is  related  to  the  attendant  free-energy- 
decrease  (  —  A  F)  of  the  system  in  the  simple  way  expressed  by  the 
equation:  —  AF  =  WR  —  2(A  pv}.  It  was  also  shown  in  Art.  125 


FREE  EXEKGY  OF  ISOTHERMAL  CHANGES  181 

that,  in  order  to  determine  this  maximum  work  or  this  free- 
energy-decrease,  the  change  in  state  must  be  made  to  take  place 
reversibly. 

It  was  shown  in  Art.  126  that  the  free-energy-decrease  attending 
the  change  in  the  pressure  from  pi  to  p3  on  any  substance  whose  vol- 
ume is  v  at  the  pressure  p  is  given  in  general  by  the  integral 
of  v  dp  from  p2  to  p1  ;  and  that  it  is  given  by  the  expression 
—  AF  =  NRT  Iog(p1/p2)  when  the  pressure-volume  relations  of 
the  substance  are  those  of  a  perfect  gas  within  the  pressure-interval 
involved. 

It  was  also  shown  in  Art.  126  that  the  free  energies  of  a  sub- 
stance in  two  different  phases  are  the  same  when  the  pressure  or 
concentration  is  that  at  which  equilibrium  prevails. 

It  was  shown  in  Art.  127  that  the  free-energy-decrease  attending 
the  transfer  at  the  temperature  T  of  N  mols  of  a  substance  from  an 
infinite  quantity  of  a  solution  in  which  its  vapor-pressure  is  pa  and 
its  concentration  is  c1  into  an  infinite  quantity  of  another  solution 
in  which  its  vapor-pressure  is  p2  and  its  concentration  C2  is  given  by 
the  expressions  : 

-AF  =  NRT  log-?!;  and  —  &F  =  NRT  log^l. 


The  first  of  these  expressions  holds  true  whatever  the  concentrations 
of  the  substance,  provided  its  vapor  conforms  to  the  perfect-gas  law  ; 
the  second  holds  true  only  when  the  concentrations  are  so  small 
that  the  solute  conforms  to  the  laws  of  perfect  solutions. 

148.  The  Free  Energies  of  Substances  in  Their  Different  Physical 
States.  —  The  principles  summarized  in  Art.  147  may  be  applied  to 
the  determination  of  the  free  energies  of  a  substance  in  its  different 
physical  states,  as  illustrated  by  the  following  problems. 

Pro&.  2.  Calculate  the  free  energy  at  25°,  a,  of  72(latm.)  ;  6,  of 
I2(at  1m.  in  H2O)  ;  and  c,  of  I,  (at  1m.  in  CC14).  At  25°  the  vapor- 
pressure  of  pure  iodine  is  0.305  mm.,  its  solubility  in  water  is  0.00132 
molal,  and  its  distribution-ratio  between  carbon  tetrachloride  and  water 
is  86. 

Pro  5.  3.  a.  Describe  a  process,  for  each  stage  of  which  the  free- 
energy-decrease  can  be  evaluated,  by  which  rhombic  sulphur  can  be 
converted  into  monoclinic  sulphur,  making  use  of  their  solubilities  in 
benzene.  &.  From  the  facts  that  these  solubilities  at  25°  are  18.2  g.  and 
23.2  g.  per  1.,  respectively,  and  that  the  molecular  formula  of  sulphur  ia 


182  THERMODYNAMIC  CHEMISTRY 

benzene  has  been  shown  by  molecular-weight  determinations  to  be  S8, 
calculate  the  free  energy  of  IS(monoclinic)  at  25°.  c.  Predict  from  the 
diagram  of  Art.  95  the  conditions  under  which  the  free  energies  of 
rhombic  and  monoclinic  sulphur  are  equal. 

Pro'b.  Jf. —  Determination  of  the  Free  Energy  of  a  Solute  at  High 
Concentration. —  a.  Formulate  a  numerical  expression  by  which  one 
may  calculate  the  free  energy  F2  of  lNH3(at  8.6  n.  in  H2O)  from  the 
free  energy  Ft  of  lNH3(at  0.2  n.  in  H2O)  with  the  aid  of  the  values  of 
the  distribution-ratio  of  NH3  between  CC14  and  H2O,  which  are  0.0086 
and  0.0040  when  the  NH3  concentrations  in  the  water  are  8.6  and  0.2 
normal,  respectively.  &.  State  the  principles  involved,  c.  State  how 
the  same  free-energy  quantity  might  be  obtained  from  other  measure- 
ments with  the  same  solutions. 

149.  Derivation  of  the  Free  Energies  of  Substances  from  Elec- 
tromotive Forces. — It  was  shown  in  Art.  129  that  changes  in  state 
can  often  be  caused  to  take  place  reversibly  in  voltaic  cells,  and 
that  the  free-energy-decrease  is  then  given  by  the  expression 
—  A?7  =  E  N  F,  which  is  valid  whatever  be  the  concentrations. 

Prol.  5.  Calculate  the  free  energy  of  IBTCZClatm.)  at  25°  from  the 
facts  that  at  25°  the  cell  H2(latm.),  HCl(4f.),  CZ2(latm.)  has  an 
electromotive  force  of  1.262  volts,  and  that  the  vapor-pressure  of  the 
HC1  in  its  4-formal  solution  is  0.0094  mm. 

Profe.  6.  Calculate  the  free  energy  of  lAgCl  at  25°  from  the  specific 
electrode-potentials. 

Prob.  7.  a.  Formulate  the  free-energy  equation  which  can  be  derived 
from  the  specific  electrode-potential  of  lead,  which  is  -f-  0.12  volt  at  25°. 
6.  State  what  conclusion  as  to  the  free  energy  of  lead-ion  can  be  drawn 
from  this  value  under  the  conventions  that  have  been  adopted. 

Prob.  8.  Calculate  the  free  energy  at  25°,  a,  of  lCl-(at  1m.),  and 
6,  of  lOH-(at  1m.). 

150.  The  Free-Energy-Decrease  attending  Isothermal  Chemical 
Changes  between  Gaseous  Substances  in  Relation  to  Their  Equilib- 
rium Conditions. — It  has  already  been  seen  in  Art.  129  that  chemical 
changes  can  often  be  made  to  take  place  reversibly  in  voltaic  cells. 
It  will  now  be  shown  that  there  is  another  kind  of  reversible  process 
by  which  they  can  be  brought  about.  This  kind  of  process,  which 
involves  only  mechanical  work,  leads  to  a  relation  between  the 
maximum  work  or  free-energy-decrease  attending  the  chemical 
Change  and  its  equilibrium-conditions. 


FREE  ENERGY  OF  ISOTHERMAL  CHANGES 


183 


Consider  any  chemical  reaction  aA  +  &B  . .  =  eE  +  /F  . . 
between  the  gaseous  substances  A,  B, . .  E,  F, . .  ;  and  consider  a 
change  in  state  which  consists  in  the  conversion  at  the  temperature 
T  of  a  mols  of  A,  ~b  mols  of  B, . .  at  pressures  p  ',  p  ', . .  into  e  mols 

A  B 

of  E  and  /  mols  of  F, . .  at  pressures  p^',  Pf'  •• 

To  carry  out  this  change  in  state  by  a  reversible  process  we  make 
use  of  an  apparatus  like  that  shown  in  Fig.  11.  This  apparatus  con- 


1 

1 

1 

GAS 

1 

GAS 

GAS 

A 

GAS 
B 

E 

F 

EQUILIBRIUM  MIXTURE  OF  A,  B,  E,  and  F 


Fig.  11 

sists  of  a  reservoir  and  of  a  number  of  cylinders  which  communicate 
with  it  through  walls,  each  of  which  is  permeable  for  only  one  of 
the  gases.  These  walls  can  be  replaced  at  will  by  impermeable  ones. 
The  cylinders,  which  serve  to  hold  the  separate  gases,  are  provided 
with  weighted  frictionless  pistons.  The  reservoir  contains  a  mixture 
of  the  gases  at  any  partial  pressures  p  ,  p  , . .  p  ,  p  , . .  at  which 

A  B  E  F 

the  chemical  change  is  in  equilibrium. 

Starting  now  with  a  mols  of  A,  ft  mols  of  B, . .  at  pressures  p  ', 
p  ' . .  in  their  respective  cylinders,  consider  the  following  process  to 

B 

be  carried  out  at  the  temperature  T  under  reversible  conditions 
(always  keeping  the  external  pressure  on  any  piston  equal  within 
an  infinitesimal  amount  to  that  of  the  gas  beneath). 

1.  The  cylinders  being  closed  temporarily  by  impermeable  walls, 
raise  or  lower  the  pistons  above  the  gases  A,  B, . .  till  the  pressures 
change  from  p  ',  p  ', . .  to  those,  p  ,  p  , . . ,  prevailing  in   the  equilib- 

A  B  A  B 

rium-imxture. 

2.  Replace  the  „  impermeable  walls  by   semipermeable   ones,   and 
force  the  a  mols  of  A,  b  mols  of  B, . .  into  the  reservoir,  at  the  same 
time  drawing  out   into  the  other  cylinders  at  the  pressures  p  ,  p ,, . . 


184  THERMODYNAMIC  CHEMISTRY 

prevailing  in  the  equilibrium-mixture  the  e  mols  of  E,  /  mols  of 
F, . .  which  must  form  spontaneously  out  of  A,  B, . .  in  order  that 
the  equilibrium  in  the  mixture  may  be  maintained. 

3.  Replace  the  semipermeable  walls  under  the  gases  E,  F, . .  by 
impermeable  ones,  and  raise  or  lower  the  pistons  above  the  gases 
E,  F, . .  till  the  pressures  change  from  the  equilibrium-pressures 
p  ,  p  , . .  to  the  final  pressures  p^,  p  ' . . 

By  formulating,  as  is  done  in  Prob.  9,  expressions  for  the  free- 
energy-decrease  attending  each  of  the  steps  in  this  process  and  sum- 
ming up  the  three  values,  there  is  obtained  in  the  case  that  all  the 
pressures  involved  are  so  small  that  they  conform  substantially  to 
the  perfect-gas  law  the  following  expression  for  the  free-energy- 
decrep.se  attending  a  chemical  change  between  gaseous  substances: 

- &F  =  RT 


Prob.  9. — Derivation  and  Significance  of  the  Free-Energy  Equation. 
— a.  Formulate  expressions  for  the  free-energy-decrease  attending  each 
of  the  three  steps  in  the  process  described  in  the  preceding  text.  &.  By 
combining  these  expressions  derive  the  free-energy  equation  there 
given,  c.  State  explicitly  the  change  in  state  to  which  this  equation 
applies  and  the  difference  in  the  significance  of  the  pressures  in  the 
fwo  logarithmic  terms ;  and  name  a  familiar  quantity  that  may  be 
substituted  in  one  of  these  terms,  d.  Give  the  form  which  the  equation 
assumes  when  the  initial  and  final  partial  pressures  are  all  unity. 

Prob.  10. —  Numerical  Applications  of  the  Free-Energy  Equation. — 
a.  Calculate  the  free-energy-decrease  in  calories  attending  the  change 
2#2(0.1atm.)  4-  O2(0.5atm.)  =  2#2O(latm.)  at  2000°  from  the  fact 
that  water-vapor  at  2000°  and  1  atm.  is  1.85%  dissociated  into  hydrogen 
and  oxygen.  b.  Calculate  also  the  free  energy  at  2000°  of  177,0(1  atm.), 
as  defined  in  Art.  146. 

Prob.  11. —  Derivation  of  the  Mass-Action  Law. — Derive  the  mass- 
action  law  expressed  in  terms  of  pressures  by  considering  in  the  deriva- 
tion of  the  free-energy  equation  that  the  same  change  in  state  is 
brought  about  by  two  reversible  processes  of  the  type  described  in 
the  text  differing  in  the  respect  that  different  equilibrium-mixtures  are 
contained  in  the  reservoir. 

Prob.  12. —  Relation  of  the  Free-Energy-Change  to  the  Tendency  of 
the  Chemical  Change  to  Take  Place. —  a.  Show  from  the  free-energy 
equation  that  the  substances  involved  in  a  chemical  change  are  in  equi- 


FREE  ENERGY  OF  ISOTHERMAL  CHANCES  185 

librium  when  their  partial  pressures  are  such  that  the  free-energy-de- 
crease would  be  zero  if  the  change  took  place,  b.  Show  that  in  any  given 
mixture  the  change  tends  to  take  place  in  that  direction  in  which  it  is 
attended  by  a  decrease  in  the  free  energy. 

The  conclusion  reached  in  the  last  problem  is  another  example 
(see  Arts.  126  and  139)  of  the  following  general  principle:  when 
the  conditions  of  temperature  and  of  pressure  or  concentration  are 
such  that  a  substance  or  a  group  of  substances  may  undergo  a 
change  in  state  without  any  attendant  change  in  free  energy,  the 
two  states  are  in  equilibrium  with  each  other.  More  briefly  ex- 
pressed, the  condition  —  A^  =  0  is  a  criterion  of  equilibrium. 

151.  The  Tree-Energy  Equation  for  Chemical  Changes  between 
Solutes  at  Small  Concentrations. — It  can  be  shown,  as  is  done  in 
Prob.  13,  that  the  free-energy-decrease  attending  at  the  tempera- 
ture T  the  chemical  change: 

oA(at  CA')  -f  bB(at  CB')  ..  =  eE(at  c/)  -f  /F(at  c/)  . . , 
where    A,  B, . .  E,   F, . .   are    solutes    at    the    small    concentrations 
c  ',  c  ', . .  c  ',  c  ' . . ,  is  given  by  the  equation : 

(,,  e  „  /  r  'e    r  'f       \ 

log  \\^- log  B,B  V  • 
c«cBb..  cAac9b../ 

In  this  equation,  which  is  obviously  closely  analogous  to  that  for 
gaseous  substances,  the  quantities  c  ,  c  , . .  c  ,  c  , . .  are  any  small 

A          B  E  F 

concentrations  at  which  the  substances  are  in  equilibrium. 

Prob.  18.  Derive  the  free-energy  equation  given  in  the  preceding 
text  by  considering  that  all  the  solutes  are  volatile  and  that  the  change 
in  state  is  brought  about  by  vaporizing  A,  B, . .  out  of  their  solutions, 
converting  them  into  E,  F, . .  in  the  gaseous  state  by  the  process  de- 
scribed in  the  text  of  Art.  150,  and  condensing  E,  F, . .  into  their  solu- 
tions; each  of  these  changes  being  carried  out  under  equilibrium-con- 
ditions so  that  it  may  be  reversible.  Represent  by  PA',  PA, . .  PF',  PF  . . 
the  partial  vapor-pressures  corresponding  to  the  concentrations  OA', 
CA,  . .  CF',  cp, . . ,  and  assume  that  in  accordance  with  Henry's  law 
the  pressures  and  concentrations  are  proportional. 

Prob.  14.  Calculate  the  free-energy-decrease  attending  the  change 
NH4+(at  0.1  f.)  -f  ON- (at  0.1  f.)  =  NH8(at  0.1  f.)  -f-  HCN(at  0.1  f.)  at  25° 
from  the  fact  that  NH/CN-  in  0.1  formal  solution  at  25°  is  52%  ionized 
into  NH4+  and  ON-  and  41%  dissociated  into  NH,  and  HCN. 


lcS6  THERMODYNAHIC  CHEMISTRY 

Although  the  free-energy  equation  has  been  derived  above  under 
the  assumptions  that  the  solutes  are  volatile  and  that  their  concentra- 
tions and  vapor-pressures  are  so  small  that  Henry's  law  is  applicable, 
it  can  be  shown  that  this  equation  (like  equation  (3)  of  Art.  127) 
is  valid  provided  only  that  the  concentrations  are  so  small  that  the 
solutes  conform  to  the  laws  of  perfect  solutions,  for  example,  to 
the  osmotic-pressure  equation  P  =  c  R  T. 

The  validity  of  the  mass-action  law  for  solutes  at  small  concen- 
trations can  be  derived  from  this  free-energy  equation  just  as  that 
for  gases  at  small  pressures  was  derived  in  Prob.  11  from  the  free- 
energy  equation  of  the  preceding  article.  The  validity  of  the  mass- 
action  law  therefore  involves  only  that  of  the  law  of  perfect  solu- 
tions or  of  perfect  gases;  and  it  will  hold  true  when  applied  to  any 
definite  chemical  change  with  a  degree  of  exactness  corresponding 
to  that  with  which  the  substances  involved  conform  under  the  given 
conditions  of  concentration  or  pressure  to  the  law  of  perfect  solutions 
or  of  perfect  gases. 

152.  The  Free-Energy  Equations  for  Chemical  Changes  between 
Solid  Substances  and  Gaseous  or  Dissolved  Substances  at  Small 
Concentrations. — When  pure  solid  substances  are  involved  in  a 
chemical  change  with  gaseous  or  dissolved  substances  it  is  not  neces- 
sary to  include  in  the  free-energy  equation  their  pressures  or  con- 
centrations; for  these  would  obviously  have  the  same  value  (that 
of  the  vapor-pressure  or  solubility  of  the  solid)  in  the  two  logarith- 
mic terms  corresponding  to  the  equilibrium  conditions  and  the  ini- 
tial and  final  conditions,  respectively.  The  same  is  true  of  the 
pressure  of  a  pure  liquid  when  it  is  involved  in  a  chemical  change 
•with  gaseous  substances  that  are  not  much  soluble  in  it,  as  in  the 
formation  of  liquid  water  from  hydrogen  and  oxygen. 

Prob.  15.  Calculate  from  the  data  of  Prob.  47,  Art.  91,  the  free- 
energy-decrease  attending  at  25°  the  chemical  change : 

NHJ3H  =  tf#s(latm.)   4.  .ff2<Sf(l  atm.).  • 

Prob.  16.  Calculate  the  free  energy  of  !Ag20  at  25°  from  the  fact 
that  its  dissociation-pressure  at  25°  is  0.38  mm. 

Prob.  17.  Calculate  from  the  data  of  Prob.  59,  Art  93,  the  free- 
energy-decrease  attending  at  25°  the  chemical  change: 

AgSCN  4.  Br-(0.1  f.)   =  AgBr  4-  SCN-(0.1  f.). 


FREE  ENERGY  OF  ISOTHERMAL  C-HANGES  187 

153.  Calculation  of  the  Equilibrium-Constants  of  Chemical 
Changes  and  of  the  Electromotive  Forces  of  Voltaic  Cells  from  the 
Free  Energies  of  the  Substances  Involved. — When  the  values  of  the 
free  energies  of  substances  have  been  once  determined  by  the  meth- 
ods already  described,  these  values  can  be  employed,  conversely,  for 
calculating  the  equilibrium-constants  of  chemical  changes  and  the 
electromotive  force  of  voltaic  cells  in  which  the  substances  are  in- 
volved. From  this  fact  arises  the  great  importance,  referred  to  in 
Art.  145,  of  a  systematic  knowledge  of  free-energy  values. 

Prob.  18.  a.  Calculate  from  the  free-energy  values  already  consid- 
ered the  equilibrium-constant  at  25°  of  the  reaction: 

Ag20  -f-  H20  -f  201-  =  2AgCl  +  2OH-. 

&.  State  what  mixture  would  finally  result  if  !Ag20  were  treated  at  25° 
with  1  1.  of  0.1  normal  NaCl  solution. 

Prob.  19.  a.  Calculate  from  the  free-energy  values  the  equilibrium- 
constant  of  the  reaction  4H Cl  -f-  O2  =  2C7a  -f  2H,O  at  25°.  &.  Calculate 
the  mol-fractions  of  chlorine  and  oxygen  in  the  gas  which  at  1  atm.  and 
25°  would  escape  from  a  4-formal  HOI  solution  if  oxygen  at  1  atm.  were 
passed  through  it  in  contact  with  a  catalytic  agent  so  that  equilibrium 
was  established.  At  25°  in  4-formal  HOI  solution  the  vapor-pressures 
of  the  water  and  HC1  are  19.6  mm.  and  0.0094  mm.,  respectively. 

Prob.  20.  Calculate  from  the  free-energy  values  and  the  other  data 
needed  the  electromotive  forces  at  25°  of  the  following  cells: 

a.  #2(latm.),  H,SO4 (0.01  f. ) ,  O2(latm.). 

b.  C^datm.),  HCl(4f.),  Oa(latm.). 


188  THERMODYNAMIC  CHEMISTRY 


THE    FUNDAMENTAL    SECOND-LAW   EQUATION 

154.  Derivation  from  the  Second  Law  of  Energetics  of  an  Ex- 
pression for  the  Quantity  of  Work  that  can  be  produced  when  a 
Quantity  of  Heat  passes  from  One  Temperature  to  Another. —  The 

remainder  of  this  chapter  is  devoted  to  a  consideration  of  the  effect 
of  temperature  on  the  free-energy-changes  that  attend  isothermal 
changes  in  state.  To  this  effect  is  closely  related,  in  virtue  of  the 
free-energy  relations  already  derived,  the  effects  of  temperature  on 
the  electromotive  force  of  voltaic  cells  and  on  the  equilibrium  of 
chemical  changes.  But  before  these  effects  can  be  properly  consid- 
ered the  more  general  thermodynamic  relation  referred  to  in  the 
title  of  this  article  must  be  known.  This  will  now  be  derived. 

It  has  been  already  stated  (in  Art.  124)  that,  when  a  quantity  of 
heat  is  transformed  into  work  by  a  cyclical  process  (that  is,  by  a 
process  in  which  the  system  undergoes  no  permanent  change  in 
state),  an  additional  quantity  of  heat  is  always  taken  up  from 
surroundings  at  a  higher  temperature  and  given  out  to  surroundings 
at  a  lower  temperature.  That  is  to  say,  even  when  a  difference  of 
temperature  exists,  only  a  fraction  of  the  heat  taken  up  by  the  sys- 
tem from  the  warmer  surroundings  can  be  transformed  into  work. 
Important  questions  at  once  arise  as  to  what  determines  the  fraction 
that  can  be  so  transformed — as  to  whether  it  is  dependent  on  the 
nature  of  the  process  employed,  and  how  it  varies  with  the 
temperatures. 

In  order  to  determine  whether  the  quantity  of  work  that  can  be 
produced  when  any  definite  quantity  of  heat  is  transferred  by  a 
cyclical  process  from  a  higher  to  a  lower  temperature  is  dependent 
upon  the  nature  of  the  system  emp^yed  for  the  transformation,  or 
upon  the  way  in  which  the  transformation  is  carried  out,  let  us 
assume  that  two  different  reversible  cycMcal  processes,  carried  out 
with  different  systems  or  in  a  different  way  with  the  same  system, 
could  produce  two  unequal  quantities  of  work  by  transferring  an 
equal  quantity  of  heat  from  a  higher  to  a  lower  temperature.  Let 
us  then  cause  the  process  that  produces  the  larger  quantity  of  work 
W"  to  take  place  in  such  a  way  that  it  takes  up  a  quantity  of 
heat  Q1  at  the  higher  temperature  T19  transfers  a  part  of  it  Q2  to 


THE  SECOND-LAW  EQUATION  189 

the  lower  temperature  T2,  and  transforms  the  remainder  into  work 
W";  and  let  us  cause  the  other  process,  which  in  transferring  the 
same  quantity  of  heat  Q2  from  T1  to  T2  produces  the  smaller  amount 
of  work  W,  to  take  place  in  the  reverse  direction — that  is,  so  that 
it  takes  up  the  heat  Q2  transferred  by  the  former  process  to  the 
lower  temperature,  and  raises  it  to  the  higher  temperature  by  ex- 
pending the  required  amount  of  work  W.  It  is  then  evident  that  the 
net  result  of  these  operations  would  be  the  production  of  a  quantity 
of  work  W"  —  W  from  an  equivalent  quantity  of  heat  without  any 
other  change  having  been  brought  about  either  in  the  systems  or 
in  the  surroundings.  Since  this  is  contrary  to  the  fundamental 
statement  of  the  Second  Law,  the  supposition  made  that  the  two 
processes  produce  unequal  quantities  of  work  is  untenable.  This 
important  conclusion  may  be  explicitly  stated  as  follows:  the  maxi- 
mum amount  of  work  which  can  be  produced  when  a  definite  quan- 
tity of  heat  is  transferred  from  one  temperature  to  another  by  any 
process  in  which  the  system  employed  undergoes  no  permanent 
change  in  state  is  not  dependent  on  the  nature  of  the  process. 

By  the  conclusion  just  reached  the  determination  of  the  relation 
between  the  temperatures  and  the  proportion  of  heat  transformable 
into  work  is  greatly  facilitated;  for  evidently  it  is  now  only  neces- 
sary to  determine  what  that  relation  is  for  a  single  reversible  cyclical 
process.  Such  a  process  is  considered  in  the  following  problem. 

Pro  6.  21.  With  N  mols  of  a  perfect  gas  contained  in  a  cylinder 
closed  with  a  weighted  frictionless  piston  and  having  a  volume  t?,  and 
temperature  T  the  following  process,  consisting  of  four  distinct  parts, 
is  carried  out  reversibly:  (1)  The  gas  is  placed  in  a  large  heat-reser- 
voir at  the  temperature  T ;  and,  by  gradually  diminishing  the  weight 
on  the  piston,  it  is  caused  to  expand  until  its  volume  becomes  v2. 

(2)  The  piston  is  fixed  so  that  the  volume  must  remain  constant,  and 
the  gas  is  placed  in  a  large  heat-reservoir  at  a  temperature  T  _j_  dT. 

(3)  Keeping  the  gas  in  the  heat-reservoir  at  the  temperature  T  -j-  dT, 
it  is  compressed  by  releasing  the  piston  and  gradually  increasing  the 
weight  upon  it  until  the  volume  v2  of  the  gas  has  been  restored  to  its 
original  value  vt.    (4)  The  piston  is  again  fixed  so  as  to  keep  the  vol- 
ume constant,  and  the  gas  is  placed  in  a  heat-reservoir  at  the  tem- 
perature T.     a.    Formulate  an  expression  for  the  quantities  of  work 
Wj,  Wv . .  produced  in  the  surroundings  and  for  the  quantities  of  heat 
Qj,  Q2, . .  withdrawn  from  them  in  the  separate  steps  of  this  process. 


190  THERMODYNAMIC  CHEMISTRY 

6.  Formulate  a  relation  between  the  quantity  of  work  2  W  produced 
in  the  whole  process  and  the  quantity  of  heat  Q  withdrawn  from  the 
reservoir  at  the  temperature  T. 

Since  it  has  been  shown  that  the  Second  Law  requires  that  the 
same  quantity  of  work  be  produced  when  a  definite  quantity  of  heat 
is  transferred  by  any  reversible  cyclical  process  whatever  from  one 
definite  temperature  to  another,  it  is  evident  that  the  equation 
derived  in  the  preceding  problem  for  one  such  process  is  an  exact 
expression  of  the  Second  Law  for  every  such  process.  This  equation 
therefore  expresses  one  of  the  fundamental  principles  of  physical 
science. 

The  equation  just  derived,  which  will  be  called  simply  the  sec- 
ond-law equation,  may,  to  show  its  significance  most  clearly,  be 
written  in  the  form: 


-         ~. 

In  this  equation  2  W  denotes  the  algebraic  sum  of  all  the  quantities 
of  work  produced  in  any  reversible  cyclical  process  taking  place  at 
two  temperatures  T  and  T  -f-  dT  in  which  the  quantity  of  heat  Q 
is  withdrawn  from  the  surroundings  at  the  temperature  T;  the 
quantity  of  this  heat  not  transformed  into  work  being  imparted  to 
the  surroundings  at  T  -f-  dT.  The  work-quantity  2  W  is  obviously 
infinitesimal  in  correspondence  with  the  infinitesimal  temperature- 
difference  dT. 

Pro  6.  22.  Show  that  the  second-law  equation  leads  to  the  following 
special  conclusions:  a.  When  there  is  no  difference  of  temperature  in 
the  surroundings  heat  cannot  be  transformed  at  all  into  work  by  any 
cyclical  process.  5.  In  order  to  carry  heat  from  a  lower  to  a  higher 
temperature  work  must  be  withdrawn  from  the  surroundings,  c.  The 
fraction  of  the  heat  transformable  into  work  for  a  given  difference  in 
temperature  is  greater  the  lower  the  temperature,  d.  When  the  higher 
temperature  is  only  infinitesimally  greater  than  the  absolute  zero  heat 
can  be  completely  transformed  into  work. 


EFFECT  OF  TEMPERATURE  O.V  EQUILIBRIUM  101 

EFFECT    OF  TEMPERATURE   ON    CHEMICAL   EQUILIBRIUM    AND 
ELECTROMOTIVE    FORCE 

155.  Effect  of  Temperature  on  the  Pressure  at  which  the  Phases 
of  Univariant  Systems  are  in  Equilibrium.  —  An  important  relation 
derivable  from  the  second  law  of  energetics  is  that  named  in  the  title 
of  this  article.  It  is  expressed  by  the  following  equation,  known  as 
the  Clapeyron  equation: 

dp        J^  A# 

dT  ~~  T   Av' 

In  this  equation  dp  denotes  the  increase  produced  by  a  temperature- 
increase  dT  in  the  pressure  at  which  the  phases  of  a  univariant  sys- 
tem (like  one  consisting  of  ice  and  water,  or  of  CaC03,  CaO,  and  (702) 
are  in  equilibrium  at  the  temperature  T,  and  A£f  and  Av  denote  the 
increases  in  heat-content  and  in  volume  which  attend  the  conversion 
at  the  temperature  T  and  at  the  equilibrium-pressure  p  of  any  defi- 
nite quantity  of  one  phase  or  set  of  phases  into  another  phase  or  set 
of  phases. 

This  equation  can  be  derived  from  the  second-law  equation  in  the 
way  shown  in  the  following  problem. 

Pro 6.  23.  Derive  the  Clapeyron  equation  from  the  second-law  equa- 
tion by  considering  the  quantities  of  work  and  heat  involved  in  the  fol- 
lowing cyclical  process :  Starting  with  some  definite  quantity  of  a  sub- 
stance (for  example,  1H2O)  existing  in  a  phase  (for  example,  liquid 
water)  which  is  in  equilibrium  with  a  second  phase  (for  example,  ice) 
at  the  temperature  T  and  pressure  p,  cause  it  to  pass  under  these  equilib- 
rium conditions  from  the  first  phase  in  which  its  volume  is  v±  into  the 
second  phase  in  which  its  volume  is  vz;  then  heat  the  second  phase  to 
T  -f-  dT,  whereby  the  pressure  becomes  p  -j-  dp  and  the  volume  v2  -j-  dv2 ; 
now  cause  it  to  go  over  into  the  first  phase  at  T  -|-  dT  under  the  equilib- 
rium pressure  p  -j-  dp,  whereby  the  volume  becomes  v:  -[-  <fi\;  and 
finally  cool  this  first  phase  to  T,  whereby  it  reverts  to  its  original  con- 
dition. 

In  applications  of  the  Clapeyron  equation  a  definite  change  in 
state  should  first  be  formulated,  and  then  AJ?  and  Av  should  be  evalu- 
ated in  accordance  with  it,  expressing  the  heat-quantity  Afl"  and  the 
work-quantity  dp  .  Av  in  the  same  units.  When  one  of  the  phases  in- 
volved is  a  gas  at  small  pressure,  Av  may  be  determined  by  neglecting 
the  volume  of  the  solid  or  liquid  phases  and  expressing  the  volume  of 


192  THERMODYXAMIC  CHEMISTRY 

the  gaseous  phase  in  terms  of  its  temperature  and  pressure  with  the 
aid  of  the  perfect-gas  law.  Applications  of  the  equation  to  systems  of 
this  kind  (those  involving  liquid  and  gaseous  phases)  were  considered 
in  Art.  22.  Other  applications  are  illustrated  by  the  following  prob- 
lems. 

Profr.  24.  Derive  from  the  Clapeyron  equation  a  principle  expressing 
the  direction  of  the  effect  of  pressure  on  the  melting-point  of  solid  sub- 
stances, taking  into  account  the  fact  that  fusion  is  always  attended  by 
an  absorption  of  heat. 

Pro  6.  25.  Calculate  the  variation  per  atmosphere  of  the  melting- 
point  of  ice.  At  0°  and  1  atm.  its  density  is  0.917  and  the  heat  of  fusion 
of  1  g.  is  79.7  cal. 

Pro  6.  26.  a.  Calculate  the  variation  per  atmosphere  of  the  transi- 
tion-temperature (95.0°)  of  rhombic  into  monoclinic  sulphur  from  the 
densities  of  the  two  forms,  which  are  2.07  and  1.96  respectively,  and 
from  the  fact  that  the  transition  of  1  at.  wt.  of  sulphur  from  the 
rhombic  into  the  monoclinic  form  is  attended  at  95°  by  an  absorption 
of  105  cal.  &.  What  does  this  show  as  to  the  slope  of  the  line  BE  in  the 
sulphur  diagram  shown  in  Fig.  7,  Art.  95  ?  and  what  conclusion  as  to  the 
slope  of  the  line  CF  can  be  drawn  by  considering  the  value  of  the 
density  of  liquid  sulphur,  which  is  1.81  at  115°  ? 

Pro&.  27.  With  the  aid  of  the  temperature-vapor-pressure  curves  of 
Fig.  8  in  Art.  101  derive  approximate  values  of  the  heat-effect  attending 
the  reaction  Na2HP047H20  4-  5H2O  =  Na2HP0412H20  at  25°. 

In  order  to  integrate  the  Clapeyron  equation  exactly,  A£f  and  Av 
must  be  expressed  as  functions  of  the  equilibrium-temperature  or  of 
the  equilibrium-pressure.  When  the  system  consists  of  only  solid  and 
liquid  phases  and  when  only  moderate  changes  of  pressure  (such  as 
20  atm.)  are  involved,  AJ?,  Av,  and  T  in  the  second  member  of  the 
equation  may  be  considered  constant  in  approximate  calculations. 
When  one  of  the  phases  consists  of  a  perfect  gas,  Av  may  be  expressed 
as  a  temperature-function  by  means  of  the  perfect-gas  equation.  The 
heat-quantity  Afi"  may  always  be  so  expressed  in  terms  of  the  heat- 
capacities  of  the  substances  involved,  in  the  way  described  in  Art.  115. 

Pro 6.  28.  Calculate  the  melting-point  of  ice  at  11  atm.  with  the  aid 
of  the  data  of  Prob.  25. 

Prol).  29.  State  what  heat-quantity  can  be  calculated  from  the  dis- 
sociation-pressures of  NH4SH  into  NH3  and  Ht8,  which  are  500  mm.  at 
25°  and  182  mm.  at  10°  ;  and  formulate  a  numerical  expression  by 
which  it  can  be  calculated,  assuming  it  to  be  constant  through  the  tem- 
perature interval  involved. 


EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM  193 

Pro&.  30.  Calculate  the  vapor-pressure  of  solid  iodine  at  25°  from 
the  following  data :  its  vapor-pressure  at  100°  is  47.5  mm.,  its  molal 
heat  of  vaporization  at  100°  is  14.600  cal.,  its  atomic  heat-capacity  is 
6.7  between  25°  and  100°,  and  the  molal  heat-capacity  of  its  vapor  at 
constant  pressure  is  7.8  between  those  temperatures. 

The  derivation  of  the  Clapeyron  equation  shows  that  it  determines 
the  equilibrium-conditions  of  any  type  of  system  in  which  an  iso- 
thermal change  in  state  can  take  place  under  a  constant  equilibrium- 
pressure  which  is  determined  only  by  the  temperature;  for  evidently 
in  all  such  cases,  and  only  in  such  cases,  will  the  work-quantities  in- 
volved in  the  cyclical  process  by  which  the  equation  was  derived  have 
the  found  values.  In  other  words,  the  equation  is  applicable  to  all 
systems,  and  only  to  systems,  of  the  univariant  type  or  of  a  type  which 
has  become  in  effect  univariant  by  the  specification  that  the  composi- 
tion of  the  phases  present  shall  remain  constant  when  the  isothermal 
change  in  state  takes  place  and  when  the  temperature  of  the  system 
is  varied.  Cases  of  the  latter  type  are  illustrated  by  the  following 
problem. 

Prob.  31.  a.  In  applying  the  Clapeyron  equation  to  determine  the 
effect  of  temperature  on  the  vapor-pressure  of  a  10%  NaCl  solution, 
specify  the  change  in  state  to  which  the  quantities  AH  and  Av  would 
correspond.  6.  State  what  equilibrium  could  be  studied  by  applying  the 
Clapeyron  equation  to  a  system  consisting  of  solid  sodium  chloride  and 
its  saturated  solution,  and  specify  the  change  in  state  to  which  the 
quantities  AH  and  Av  would  correspond. 

156.  Fundamental  Equation  expressing  the  Effect  of  Temperature 
on  the  Free-Energy-Changes  attending  Isothermal  Changes  in  State. 

Prob.  32.  A  certain  reversible  cyclical  process  involves  the  follow- 
ing steps:  (1)  Any  change  in  state  of  any  system  at  the  temperature  T, 
by  which  a  quantity  of  work  W  is  produced;  (2)  a  change  in  the 
temperature  of  the  system  at  constant  volume  from  T  to  T  -\-  dT; 

(3)  a  change  in  the  state  of  the  system  at  the  temperature  T  -\-  dT 
which  is  the  reverse  of  the  change  in  state  in  the  first  step,  this  change 
being  attended  by  a  production  of  a  quantity  of  work  — (W  +  dW)  ; 

(4)  a  change  in  the  temperature  of  the  system  at  constant  volume  from 
T  -f  d T  to  T.    a.   Find  an  expression  for  2  W  for  this  process.     &.    By 
substituting  it  in  the  second-law  equation  and  by  making  other  appro- 
priate substitutions  show  that  the  change  d( — AA)  with  the  temper- 
ature of  the  decrease  in  work-content  attending  any  isothermal  change 
in  state  is  expressed  by  the  equation 

(d  —  AA)_  Atf—  AA 
dT      ~~        T      ' 


194  THERMO  DYNAMIC  CHEMISTRY 

in  which    At/  represents  the  change  in  the  energy-content  of  the  system 
attending  the  change  in  its  state  at  the  temperature  T. 

Prob.  33.  A  certain  reversible  cyclical  process  involves  the  same 
steps  as  the  process  described  in  Prob.  23  except  that  the  system  during 
the  changes  in  temperature  in  steps  (2)  and  (4)  is  kept  at  constant 
pressure,  instead  of  at  constant  volume,  a.  Find  an  expression  for 
2W  for  this  process,  representing  the  pressures  and  volumes  of  the 
system  at  the  beginning  of  each  of  the  four  steps  by  (1)  px  and  vl; 
(2)  p2  and  V2;  (3)  p2  and  v2-{-dv2;  and  (4)  p1  and  v1-\-dv1.  6.  By 
substituting  this  expression  in  the  second-law  equation  and  making 
other  appropriate  substitutions  based  on  the  definitions  of  —  AF  and 
Aff  given  in  Arts.  125  and  112,  show  that  the  change  d(—  AF)  with 
the  temperature  of  the  free-energy-decrease  attending  any  isothermal 
change  in  state  is  expressed  by  the  equation 

d(  —  AF)       Aff—  AF 


in  which    A#  represents  the  change  in  the  heat-content  of  the  system 
attending  the  change  in  its  state  at  the  temperature  T. 

The  equations  derived  in  the  preceding  problems: 


d(-AA)      AC7-AA  d(-AF)       A#  -  AF 

_  r=  --      and  —  =  - 

dT  T  dT  T 

are  fundamental  expressions  of  the  Second  Law  in  the  form  most 
suitable  for  physico-chemical  applications.  It  is  important  fully  to 
appreciate  the  significance  of  the  quantities  occurring  in  them  and 
the  condition  under  which  each  equation  is  applicable.  The  quan- 
tities —  A  A  and  —  AF  denote  the  decreases  in  the  work-content  and 
in  the  free-energy-content  of  the  system  which  attend  any  isothermal 
change  in  its  state  at  the  temperature  T;  and  A  U  and  A  H  denote 
the  accompanying  increases  in  its  energy-content  and  heat-content. 
The  differential  quantities  d(—  A  A)  and  d(—  AF)  signify  that 
when  the  same  change  in  state  takes  place  at  T  -\-  dT,  instead  of 
at  T,  it  is  attended  by  a  work-content-decrease  of  —  (  A  A  -f-  d(  A  A)) 
and  by  a  free-energy-decrease  of  —  (  AF  -}-  d(AF)),  instead  of  one  of 
—  AA  and  of  —  Af.  The  work-content  equation  is  applicable  to  cases 
where  the  initial  volume,  and  also  the  final  volume,  of  the  system 
is  the  same  at  the  two  temperatures;  and  the  free-energy  equation 
is  applicable  to  cases  where  the  initial  pressure,  and  also  the  final 
pressure,  is  the  same  at  the  two  temperatures. 


EFFECT  OF  TEMPERATURE  OX  EQUILIBRIUM  195 

Throughout  the  following  considerations  only  the  free-energy 
equation  will  be  employed.  For  purposes  of  integration  this  equation 
is  more  conveniently  written  in  the  following  forms: 


-AF2       - 

dT. 


This  equation  in  any  of  its  forms  will  be  called  the  second-law  free- 
energy  equation. 

Prob.  34.  Show  that  the  two  differential  forms  of  the  free-energy 
equation  are  identical  by  carrying  out  in  the  first  member  of  the  second 
one  the  indicated  differentiation  and  by  making  other  simple  trans- 
formations. 

157.  The  Effect  of  Temperature  on  the  Electromotive  Force  of 
Voltaic  Cells.  —  By  substituting  in  the  sectfnd-law  free-energy  equa- 
tion just  derived  the  expressions  previously  obtained  (in  Arts.  126, 
127,  129,  and  150-152)  for  the  free-energy-decrease  attending  different 
changes  in  state  more  specific  relations  expressing  the  effect  of  tem- 
perature result.  Thus  by  substituting  for  —  AF  the  value  E  N  F,  the 
following  expression,  known  as  the  Gibbs-HelmlioUz  equation,  for  the 
effect  of  temperature  on  the  electromotive  force  of  voltaic  cells  is 
obtained  : 


. 

In  this  equation  A-Z?  denotes  the  increase  in  heat-content  which 
attends  the  change  in  state  that  takes  place  when  N  faradays  of 
electricity  flow  through  a  cell  of  electromotive  force  E  containing 
infinite  quantities  of  the  constituent  substances. 

Prob.  35.  Show  from  the  Gibbs-Helmholtz  equation  under  what  con- 
ditions the  electromotive  force  of  a  cell,  a,  is  independent  .of  the  tem- 
perature; b,  is  proportional  to  the  absolute  temperature;  c,  increases 
with  rising  temperature;  d,  decreases  with  rising  temperature. 

Prob.  36.  a.  Calculate  the  temperature-coefficient  at  25°  of  the 
electromotive  force  of  the  cell  H2(l  atm.),  H2SO<(0.01  f.),  O2(l  atm.). 
b.  Calculate  the  electromotive  force  of  this  cell  at  0°,  assuming  that 
the  heat-effect  attending  the  change  in  state  does  not  vary  appreciably 
between  0  and  25°. 

Prob.  37.  Calculate  the  temperature-coefficient  at  25°  of  the  electro- 
motive force  (1.262  volts)  of  the  cell  #3(1  atm.),  HC1(4  f.),  C72(l  atm.) 


196  THERMODYNAMIC  CHEMISTRY 

from   the  heat  of  formation  of  IHCl  (—22,000  cal.)    and   its  heat  of 
solution   (given  in  Art.   121). 

Prol).  38.  a.  State  what  heat  data  would  be  needed  to  calculate 
accurately  the  temperature-coefficient  of  the  specific  electrode-potential 
of  zinc  at  25°.  ft.  Calculate  its  value,  referring  to  Art.  121  for  the 
heat  data. 

Prol).  39.  Calculate  the  electromotive  force  at  40°  of  the  storage- 
cell  Pb  -f  PbS04,  H2SO410H2O.  PbS04  -f  Pb02,  from  its  electromotive 
force  (2.096  volts)  at  0°  and  the  heat  data  needed.  The  heat-contents 
at  18°  of  !PbS04,  !Pb02,  and  1H2SO4  are  —216,210  cal.,  —62,900  cal.,  and 
— 192,900  cal.,  respectively.  For  the  heat  data  relating  to  sulphuric 
acid  solutions  see  Probs.  41  and  43  of  Art.  121.  Assume  that  the  change 
in  the  heat-content  of  the  cell  does  not  vary  between  0  and  40°. 

Pro 6.  40.  The  electromotive  force  at  the  temperature  t  of  the  Clark 
cell,  Zn  -f-  ZnS047H20,  ZnSO416.8H2O,  Hg2S04  -f  Hg,  has  been  very 
accurately  measured  and  found  to  be  1.4325  —  0.00119  (t  — 15)  — 
0.000007 (t  — 15) 2  volts,  a.  Calculate  the  change  in  the  heat-content 
of  the  cell  when  two  faradays  flow  through  it  at  20°.  6.  Considering  as 
in  Prob.  52,  Art.  140,  the  change  in  state  that  takes  place  in  it,  calculate 
the  change  in  the  heat-content  from  the  following  thermochemical  equa- 
tions : 

2Hg  -f  S  -f  202  =  Hg2S04  -f  175,000  cal. 

Zn  -f-  S  -f  2O2  —  ZnS04  -f-  230,090  cal. 

ZnS04  -f  7H20  =  ZnS047H20  +  22,690  cal. 

ZnS047H20  =  ZnS047H2O    (in  ZnSO416.8H2O)  —  4,700  cal. 

H2O  =  H2O(in  ZnSO416.8H,O)   -f  50  cal. 

Pro 6.  41-  The  electromotive  force  of  the  cell  Hg  -j-  Hg2Cl2, 
KCK0.01  f.)  -f  KNO,(1  f.),  KN03(1  f.)  +  KOH(0.01  f.),  Hg20  -f  Hg 
at  0°  is  0.1483  volt  and  at  18.5°  is  0.1636  volt.  Formulate  the  change  in 
state  that  takes  place  in  the  cell  when  two  faradays  pass  through  it ;  and 
calculate  the  attendant  increase  in  its  heat-content. 

158.  Comparison  of  the  Electrical  Work  Producible  by  Voltaic 
Cells  and  the  Change  in  Their  Heat-Content. —  The  two  laws  of  ther- 
modynamics do  not  show  that  there  is  any  relation  between  the  value 
of  the  decrease  of  free  energy  and  the  value  of  the  decrease  of  heat- 
content  attending  any  change  in  state;  for,  though  the  difference 
between  these  two  values  is  by  the  second-law  free-energy  equation 
brought  into  relation  with  the  temperature-coefficient  of  the  free- 
energy-decrease,  this  temperature-coefficient  may  have  any  magnitude 
whatever.  The  two  quantities,  — &F  and  — kH,  are  therefore  two 
thermodynamically  independent  constants,  related  to  the  second  law 
and  to  the  first  law  respectively,  each  of  which  is  determined  solely  by 
the  change  in  state. 


EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM  197 

Correspondingly  there  is  no  definite  relation  between  the  electrical 
work  producible  by  a  voltaic  cell  and  the  change  in  its  heat-content. 
It  is  necessary  to  emphasize  this  fact,  since  it  was  earlier  believed 
that  the  two  quantities  were  at  least  approximately  equal,  and  that 
the  electromotive  force  of  voltaic  cells  could  therefore  be  satisfactorily 
calculated  under  this  assumption.  This  erroneous  assumption,  which 
is  called  Thomson's  rule,  is  expressed  by  the  equation  —  Aff  =  E  N  F. 

Prob.  Ji2.  a.  Tabulate  beside  one  another  the  decreases  in  heat- 
content  and  in  free  energy  which  take  place  when  two  faradays  pass' 
through  each  of  the  cells  considered  in  Probs.  39,  40,  and  41.  Include  in 
the  table  also  the  cell  Cu,  CuAcalOOHjO,  PbAc2100H2O,  Pb,  which  at  0° 
has  an  electromotive  force  0.476  volt  and  which  undergoes  a  decrease 
in  heat-coiitent  of  8770  cal.  per  faraday.  &.  State  the  characteristic 
feature  of  each  of  these  four  cells  with  respect  to  the  two  energy-effects. 

Prob.  43.  Heat-Effects  Attending  Voltaic  Action.  —  a.  Formulate  the 
heat-effects  that  occur  in  the  surroundings  when  two  faradays  pass 
through  each  of  the  cells  named  in  Prob.  42,  assuming  that  the  heat- 
effect  arising  from  the  resistance  ,is  negligible.  b.  State  qualitatively 
what  temperature-change  would  occur  in  each  cell  if  it  were  heat-insu- 
lated from  the  surroundings. 

159.  Effect  of  Temperature  on  the  Equilibrium  of  Chemical 
Changes  involving  Gaseous  Substances  at  Small  Pressures.  —  By  sub- 
stituting in  the  second-law  free-energy  equation  given  at  the  end  of 
Art.  156  the  expression  derived  in  Arts.  150  and  152  for  the  free- 
energy-decrease  attending  an  isothermal  chemical  change  between 
gaseous  substances,  or  between  solid  and  gaseous  substances,  at  small 
pressures,  and  noting  that  the  initial  and  final  pressures  occurring  in 
this  expression  must  not  vary  with  the  temperature  if  the  second-law 
free-energy  equation  is  to  be  applicable,  there  is  obtained  the  follow- 
ing equation,  commonly  called  the  van't  Hoff  equation,  expressing 
the  effect  of  temperature  on  the  equilibrium-constant  K  of  the  chem- 
ical change: 

e       f 


Prob.  44.  Derive  the  van't  Hoff  equation  in  the  way  indicated  in  the 
preceding  text. 

Direction  of  the  Effect  of  Temperature.— 

Prob.  45.  Derive  from  the  van't  Hoff  equation  a  principle  expressing 
a  relation  between  the  direction  in  which  an  equilibrium  is  displaced  by 


198  THERMODYNAMIC  CHEMISTRY 

increase  in  temperature  and  the  sign  of  the  change  in  heat-content 
attending  the  reaction. 

Prob.  46.  a.  Show  how  the  equilibrium  in  a  gaseous  mixture  of  C12, 
HCl,  O2,  and  H2O  at  25°  would  be  displaced  by  increasing  the  tempera- 
ture. The  heat  of  formation  at  25°  of  IH  Cl  is  —22,000  cal.  and  that  of 
1#2O  is  —  57,800  cal.  b.  The  dissociation-pressure  of  CaC03  (and  of  all 
other  substances  which  dissociate  into  one  or  more  gaseous  products) 
increases  with  rising  temperature.  State  whether  heat  is  absorbed  or 
evolved  when  the  dissociation  takes  place  at  constant  temperature. 

In  order  to  integrate  the  van't  Hoff  equation,  the  heat-content- 
increase  A#  must  be  expressed  as  a  function  of  the  temperature. 
When  the  heat  of  the  reaction  has  been  measured  at  a  series  of  tem- 
peratures, this  function  can  be  derived  directly  from  the  results  of  the 
measurements.  In  most  cases,  however,  the  temperature-function  is 
obtained  from  a  knowledge  of  the  heat-content-increase  at  some  one 
temperature  and  of  the  heat-capacities  at  constant  pressure  of  the 
substances  involved  in  the  reaction,  with  the  aid  of  the  expression 
d(&H)  =  A  (7  .  dT,  in  which  A(7  represents  the  difference  between  the 
heat-capacity  (equal  to  (7E  +  (7F  .  .)  of  the  system  in  its  final  state 
and  its  heat-capacity  (equal  to  C  A  -\-  CB  .  .)  in  its  initial  state.  To  it 
corresponds  the  partially  integrated  equation 

f  A(7  dT, 

in  which  A7/0  is  an  integration-constant.  To  complete  the  integration 
A  C  must  evidently  be  expressed  as  a  function  of  the  temperature. 

Prob.  47.  a.  Derive  the  expression  d(AH)  •=.  AC  .  dT  as  described 
in  Art.  115.  b.  Integrate  it  for  the  case  that  AC  is  a  function  of  the  form 
AC  =  AC0  -f  aT  -\-  fiT2,  where  AC0,  a,  and  0  are  constants,  c.  Show 
how  a  corresponding  numerical  expression  for  Afl  for  the  reaction 
200  _}-  O2  =  2CO2  can  be  obtained  from  the  heat-capacity  data  given 
in  Art  118  and  the  heats  of  formation  of  ICO  and  1CO2  at  20°,  which 
are  —29,000  and  —97,000  respectively. 

Prob.  48.  Integrate  the  van't  Hoff  equation  between  the  limits  T^ 
and*  T2,  K^  and  K2,  for  the  following  cases  :  a,  when  AC  is  zero  and  there- 
fore A#  does  not  vary  with  the  temperature  ;  b,  when  AC  does  not  vary 
with  the  temperature;  c,  when  AC  varies  with  the  temperature  in  the 
way  stated  in  Prob.  47. 

In  numerical  applications  of  the  van't  Hoff  equation,  in  order  to 
guard  against  errors  in  the  sign  of  the  heat-content-increase  and  in 
its  value  with  respect  to  the  multiple  chosen,  the  reaction  under  con- 
sideration should  first  be  formulated  in  a  definite  chemical  equation, 


A/7  = 


.      EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM  199 

and  then  values  of  K,  A#,  and  A  (7  should  be  adopted  in  conformity 
with  it.  For  the  heat-capacities  of  the  more  important  gaseous  sub- 
stances and  of  elementary  solid  substances  see  Arts.  118  and  120. 

Prob.  49.  It  has  been  found  that  when  dry  air  (consisting  of  21.0 
molpercent  of  oxygen,  78.1  molpercent  of  nitrogen,  and  0.9  molpercent 
of  argon)  is  kept  at  1957°  till  equilibrium  is  reached,  4.3%  of  the  oxy- 
gen present  is  converted  into  nitric  oxide.  Calculate  the  molperceut 
that  would  be  so  converted  at  3000°.  The  formation  of  ItfO  from  its 
elements  at  20°  is  attended  by  a  heat-absorption  of  21,600  cal. 

Prob.  50.  The  dissociation-pressure  of  HgO  (into  mercury  and  oxy- 
gen gases)  at  390°  is  180  mm.  Calculate  its  dissociation-pressure  at 
480°.  The  heat  of  formation  of  IHgO  at  20°  is  —21,700  cal.  The  vapor- 
ization of  IHg  at  its  boiling-point  (357°)  absorbs  14,160  cal.  The  mean 
value  of  the  atomic  heat-capacity  of  liquid  mercury  between  20°  and 
357°  is  6.36  cal.  per  degree ;  and  the  heat-capacity  of  IHgO  is  10.87  at  all 
temperatures.  Assume  as  a  sufficient  approximation  that  the  heat- 
capacity  of  oxygen  is  constant  at  its  mean  value  between  20°  and  480°. 

Prob.  51.  Formulate  an  exact  numerical  expression  by  which  the 
dissociation  y  of  carbon  dioxide  into  carbon  monoxide  and  oxygen  at 
any  temperature  T  and  any  total  pressure  p  can  be  calculated.  Its  dis- 
sociation at  1205°  and  1  atm.  is  0.032%.  The  heats  of  formation  at  20° 
of  ICO  and  1CO2  are  —29,000  and  —97,000  cal. 

Prob.  52.  When  a  mixture  of  0.49  mol  O2  and  1.00  mol  HC1  is  kept 
at  386°  and  1  atm.  in  contact  with  solid  cuprous  chloride  (which  acts 
as  a  catalyzer)  till  equilibrium  is  reached,  80%  of  the  HC1  is  converted 
into  C12.  a.  Formulate  a  numerical  expression  for  calculating  the 
equilibrium-constant,  b.  Formulate  a  numerical  expression  for  cal- 
culating the  free-energy-decrease  attending  some  specified  change  in 
state  involving  each  of  the  substances  at  a  partial  pressure  of  1  atm. 

c.  Formulate  as  a  function  of  the  absolute  temperature  the  increase  in 
heat-content   attending  this   change  in    state,   using   the  heat-capacity 
data  given  in  Art.  118  and  the  heat-content  data  given  in  Prob.  46. 

d.  Derive  from  these  results  a  numerical  expression  for  calculating  the 
equilibrium-constant  at  25°  of  the  reaction  between  O2  -j-  4HCI  —  2C72 
-f  2H2O. 

160.  Effect  of  Temperature  on  the  Equilibrium  of  Chemical 
Changes  involving  Solutes  at  Small  Concentrations. — By  substituting 
in  the  second-law  free-energy  equation  the  logarithmic  concentration 
expression  for  the  free-energy-decrease  attending  chemical  changes 
between  solutes  at  small  concentrations  given  in  Art.  151,  and  noting 
that  the  last  term  in  that  expression  (containing  the  initial  and  final 
molal  concentrations)  does  not  vary  with  the  temperature,  we  get 


200  THERMODYNAM1C  CHEMISTRY 

again  the  van't  Hoff  equation,  now  expressed  in  terms  of  the  equi 
librium-concentrations  : 


This  equation  is  applicable  also  to  chemical  changes  between  solid 
substances  and  solutes  at  small  concentrations,  since  it  was  shown  in 
Art.  152  that  the  participation  of  solid  substances  in  the  change  does 
not  affect  the  free-energy  expression.  It  applies  also  to  changes  in 
which  the  solvent  (thus  the  water  in  aqueous  solutions)  enters  into 
reaction  with  solutes  at  small  concentrations. 

From  the  equation,  as  from'the  corresponding  one  in  terms  of 
pressures,  can  be  derived  the  important  qualitative  principle  that  the 
equilibrium  of  a  chemical  change  is  displaced  by  increase  of  temper- 
ature in  that  direction  in  which  the  reaction  is  attended  by  an  in- 
crease in  heat-content  (or  by  an  absorption  of  heat). 

The  integration  of  the  equation  evidently  involves  the  expression 
of  the  heat-content  as  a  function  of  the  temperature.  Since  in  the 
case  of  solutions  the  temperature-interval  often  is  not  large,  the  in- 
crease in  heat-content  can  frequently  be  regarded  as  constant  ;  and  it 
is  to  be  so  regarded  in  the  following  problems  unless  otherwise  stated. 
When  this  is  not  admissible  its  variation  with  the  temperature  must 
be  known.  This  must  usually  be  derived  from  direct  determinations 
of  the  heat-effects  attending  the  reaction  at  two  or  more  different 
temperatures;  for  there  is  likely  to  be  a  large  error  involved  in  cal- 
culating it  from  the  partial  heat-capacities  of  the  solutes,  since  they 
form  only  a  small  part  of  the  total  heat-capacity  of  the  solution. 

ProT).  53.  State  the  conclusions  in  regard  to  heats  of  solution  that 
can  be  drawn  from  the  facts  that  the  solubility  of  all  gaseous  substances 
is  decreased,  and  that  of  most  solid  substances  is  increased,  by  an  in- 
crease of  temperature. 

ProJ).  54'  The  neutralization  of  largely  ionized  univalent  acids  and 
bases  in  fairly  dilute  solution  has  been  found  by  direct  measurements 
at  2°,  10°,  18°,  26°,  and  34°  to  evolve  14,750  —  52*  cal.  at  the  tempera- 
ture t.  The  value  of  the  ionization-constant  of  water  at  25°  is 
0.8  x  10~"  Calculate  its  value  at  0°. 

Profc.  55.  Calculate  the  hydrolysis  at  0°  of  NH+4CN-  from  the  data 
of  Prob.  14  and  from  the  fact  that  on  mixing  at  25.00°  a  solution  con- 


EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM  201 

taming  0.2  NH3  and  1000  g.  water  with  one  containing  0.2  HCN  and 
1000  g.  water  and  bringing  the  mixture  back  to  25.00°  there  is  a  heat- 
evolution  of  152  cal.,  taking  into  account  the  fact  that  the  heat  of  ion- 
ization  of  largely  ionizing  substances  like  NH4CN  is  approximately  zero. 

Prob.  56.  The  solubility  of  silver  chloride  in  water  is  1.10  X  10~5 
formal  at  20°  and  15.2  x  10~6  formal  at  100°.  a.  State  what  heat-quan- 
tity can  be  computed  from  these  data,  and  calculate  its  value.  &.  State 
by  what  thermochemical  measurement  it  could  be  determined  experi- 
mentally. 

Prob.  57.  The  solubility  of  potassium  perchlorate  is  0.0781  normal 
at  10°  and  0.1800  formal  at  30°,  and  the  ionization  (derived  from  the 
conductance-ratio)  in  the  saturated  solution  is  0.823  at  10°  and  0.764 
at  30°.  a.  State  wrhat  heat-quantity  can  be  derived  from  these  data,  and 
compute  its  value.  b.  State  in  what  respect  this  quantity  differs  theo- 
retically from  the  heat  of  solution  (+  12,130  cal.)  which  has  been 
thermochemically  determined  by  dissolving  at  20°  1KC1O4  in  enough 
water  to  form  with  it  a  saturated  solution. 

161.  Derivation  of  Free-Energy  Values  at  One  Temperature  from 
Those  at  Another  Temperature.  —  The  second-law  free-energy  equation 


T    I         T 

has  over  the  more  specialized  Gibbs-Helmholtz  and  van't  Hoff  equa- 
tions derived  from  it  the  advantage  that  there  can  be  substituted  in 
it  values  of  the  free-energy-decrease  obtained  from  different  sources  — 
from  the  electromotive  force  of  a  cell  in  which  the  change  in  state 
under  consideration  takes  place,  from  the  equilibrium  conditions  of 
that  change,  or  from  the  free  energies  of  the  separate  substances  in- 
volved in  it.  It  thus  enables  an  equilibrium-constant  at  one  tempera- 
ture to  be  calculated  from  an  electromotive  force  at  another  tem- 
perature, or  the  reverse  ;  and  it  enables  either  of  these  to  be  calculated 
at  one  temperature  from  the  free  energies  of  the  involved  substances 
at  another  temperature,  or  the  free  energies  themselves  to  be  cal- 
culated over  from  one  temperature  to  another  —  provided  always  that 
the  necessary  heat  data  are  available.  It  differs  furthermore  from  the 
van't  Hoff  equation  in  the  respect  that  it  is  not  limited  to  small  pres- 
sures or  concentrations. 

Prob.  58.  Calculate  the  free  energy  of  IHCl  at  1  atm.  and  1200° 
and  its  percentage  decomposition  into  hydrogen  and  chlorine  at  1200° 
from  the  data  of  Probs.  5  and  37  and  the  heat-capacity  data  of  Art.  118. 


202  THERMODYNAMIC  CHEMISTRY 

Pro&.  59.  Show  how  the  equilibrium-constant  of  the  reaction 
HaS  -f  I2  —  S  (rhombic)  +  2H+I~  in  dilute  aqueous  solution  at  25° 
can  be  calculated  from  the  dissociation  at  90°  of  H2S  (into  H2  and  S), 
from  the  specific  electrode-potentials  at  25°,  and  from  such  other  data  as 
are  needed. 

Pro  6.  60.  Calculate  the  free  energy  at  25°  of  1  at.  wt.  of  monoclinic 
sulphur  from  the  facts  that  its  conversion  into  rhombic  sulphur  at  its 
transition-point  (95.0°)  is  attended  by  a  heat-evolution  of  105  cal.,  and 
that  the  atomic  heat-capacities  of  the  monoclinic  and  rhombic  forms  are 
6.0  and  5.7  cal.  per  degree.  Tabulate  beside  this  result  that  obtained  in 
Prob.  3. 


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